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Type | Label | Description |
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Statement | ||
Theorem | mat1dimmul 20101 | The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.) |
⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ({〈𝑂, 𝑋〉} (.r‘𝐴){〈𝑂, 𝑌〉}) = {〈𝑂, (𝑋(.r‘𝑅)𝑌)〉}) | ||
Theorem | mat1dimcrng 20102 | The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.) |
⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐸 ∈ 𝑉) → 𝐴 ∈ CRing) | ||
Theorem | mat1f1o 20103* | There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) | ||
Theorem | mat1rhmval 20104* | The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) | ||
Theorem | mat1rhmelval 20105* | The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) | ||
Theorem | mat1rhmcl 20106* | The value of the ring homomorphism 𝐹 is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) ∈ 𝐵) | ||
Theorem | mat1f 20107* | There is a function from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾⟶𝐵) | ||
Theorem | mat1ghm 20108* | There is a group homomorphism from the additive group of a ring to the additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 GrpHom 𝐴)) | ||
Theorem | mat1mhm 20109* | There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑀 MndHom 𝑁)) | ||
Theorem | mat1rhm 20110* | There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴)) | ||
Theorem | mat1rngiso 20111* | There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = ({𝐸} Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑂 = 〈𝐸, 𝐸〉 & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴)) | ||
Theorem | mat1ric 20112 | A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.) |
⊢ 𝐴 = ({𝐸} Mat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝑅 ≃𝑟 𝐴) | ||
According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple 𝜆 ∗ 𝐼 of the identity matrix 𝐼. Its effect on a vector is scalar multiplication by 𝜆 [see scmatscm 20138!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name. The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat 20115 and df-scmat 20116), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20126), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20145), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20148) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20146). | ||
Syntax | cdmat 20113 | Extend class notation for the algebra of diagonal matrices. |
class DMat | ||
Syntax | cscmat 20114 | Extend class notation for the algebra of scalar matrices. |
class ScMat | ||
Definition | df-dmat 20115* | Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.) |
⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) | ||
Definition | df-scmat 20116* | Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn". (Contributed by AV, 8-Dec-2019.) |
⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))}) | ||
Theorem | dmatval 20117* | The set of 𝑁 x 𝑁 diagonal matrices over (a ring) 𝑅. (Contributed by AV, 8-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) | ||
Theorem | dmatel 20118* | A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) | ||
Theorem | dmatmat 20119 | An 𝑁 x 𝑁 diagonal matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) | ||
Theorem | dmatid 20120 | The identity matrix is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝐷) | ||
Theorem | dmatelnd 20121 | An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 ) | ||
Theorem | dmatmul 20122* | The product of two diagonal matrices. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(.r‘𝐴)𝑌) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r‘𝑅)(𝑥𝑌𝑦)), 0 ))) | ||
Theorem | dmatsubcl 20123 | The difference of two diagonal matrices is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(-g‘𝐴)𝑌) ∈ 𝐷) | ||
Theorem | dmatsgrp 20124 | The set of diagonal matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴)) | ||
Theorem | dmatmulcl 20125 | The product of two diagonal matrices is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(.r‘𝐴)𝑌) ∈ 𝐷) | ||
Theorem | dmatsrng 20126 | The set of diagonal matrices is a subring of the matrix ring/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) | ||
Theorem | dmatcrng 20127 | The subring of diagonal matrices (over a commutative ring) is a commutative ring . (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐶 ∈ CRing) | ||
Theorem | dmatscmcl 20128 | The multiplication of a diagonal matrix with a scalar is a diagonal matrix. (Contributed by AV, 19-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → (𝐶 ∗ 𝑀) ∈ 𝐷) | ||
Theorem | scmatval 20129* | The set of 𝑁 x 𝑁 scalar matrices over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}) | ||
Theorem | scmatel 20130* | An 𝑁 x 𝑁 scalar matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) | ||
Theorem | scmatscmid 20131* | A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) | ||
Theorem | scmatscmide 20132 | An entry of a scalar matrix expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐶 ∗ 1 )𝐽) = if(𝐼 = 𝐽, 𝐶, 0 )) | ||
Theorem | scmatscmiddistr 20133 | Distributive law for scalar and ring multiplication for scalar matrices expressed as multiplications of a scalar with the identity matrix. (Contributed by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 )) | ||
Theorem | scmatmat 20134 | An 𝑁 x 𝑁 scalar matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) | ||
Theorem | scmate 20135* | An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) | ||
Theorem | scmatmats 20136* | The set of an 𝑁 x 𝑁 scalar matrices over the ring 𝑅 expressed as a subset of 𝑁 x 𝑁 matrices over the ring 𝑅 with certain properties for their entries. (Contributed by AV, 31-Oct-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}) | ||
Theorem | scmateALT 20137* | Alternate proof of scmate 20135: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 20136 but is longer and requires more distinct variables. (Contributed by AV, 19-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) | ||
Theorem | scmatscm 20138* | The multiplication of a matrix with a scalar matrix corresponds to a scalar multiplication. (Contributed by AV, 28-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)) | ||
Theorem | scmatid 20139 | The identity matrix is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝑆) | ||
Theorem | scmatdmat 20140 | A scalar matrix is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐷)) | ||
Theorem | scmataddcl 20141 | The sum of two scalar matrices is a scalar matrix. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatsubcl 20142 | The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(-g‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatmulcl 20143 | The product of two scalar matrices is a scalar matrix. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(.r‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatsgrp 20144 | The set of scalar matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐴)) | ||
Theorem | scmatsrng 20145 | The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴)) | ||
Theorem | scmatcrng 20146 | The subring of scalar matrices (over a commutative ring) is a commutative ring. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CRing) | ||
Theorem | scmatsgrp1 20147 | The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) | ||
Theorem | scmatsrng1 20148 | The set of scalar matrices is a subring of the ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) | ||
Theorem | smatvscl 20149 | Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 20056 analog.) (Contributed by AV, 24-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ ∗ = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑆)) → (𝐶 ∗ 𝑋) ∈ 𝑆) | ||
Theorem | scmatlss 20150 | The set of scalar matrices is a linear subspace of the matrix algebra. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (LSubSp‘𝐴)) | ||
Theorem | scmatstrbas 20151 | The set of scalar matrices is the base set of the ring of corresponding scalar matrices. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶) | ||
Theorem | scmatrhmval 20152* | The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 )) | ||
Theorem | scmatrhmcl 20153* | The value of the ring homomorphism 𝐹 is a scalar matrix. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) ∈ 𝐶) | ||
Theorem | scmatf 20154* | There is a function from a ring to any ring of scalar matrices over this ring. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾⟶𝐶) | ||
Theorem | scmatfo 20155* | There is a function from a ring onto any ring of scalar matrices over this ring. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–onto→𝐶) | ||
Theorem | scmatf1 20156* | There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1→𝐶) | ||
Theorem | scmatf1o 20157* | There is a bijection between a ring and any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1-onto→𝐶) | ||
Theorem | scmatghm 20158* | There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | ||
Theorem | scmatmhm 20159* | There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑇 = (mulGrp‘𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑀 MndHom 𝑇)) | ||
Theorem | scmatrhm 20160* | There is a ring homomorphism from a ring to the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | scmatrngiso 20161* | There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingIso 𝑆)) | ||
Theorem | scmatric 20162 | A ring is isomorphic to every ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅 ≃𝑟 𝑆) | ||
Theorem | mat0scmat 20163 | The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 20140, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) | ||
Theorem | mat1scmat 20164 | A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 20140, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ (#‘𝑁) = 1 ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐵 → 𝑀 ∈ (𝑁 ScMat 𝑅))) | ||
The module of 𝑛-dimensional "column vectors" over a ring 𝑟 is the 𝑛-dimensional free module over a ring 𝑟, which is the product of 𝑛 -many copies of the ring with componentwise addition and multiplication. Although a "column vector" could also be defined as n x 1 -matrix (according to Wikipedia "Row and column vectors", 22-Feb-2019, https://en.wikipedia.org/wiki/Row_and_column_vectors: "In linear algebra, a column vector [... ] is an m x 1 matrix, that is, a matrix consisting of a single column of m elements"), which would allow for using the matrix multiplication df-mamu 20009 for multiplying a matrix with a column vector, it seems more natural to use the definition of a free (left) module, avoiding to provide a singleton as 1-dimensional index set for the column, and to introduce a new operator df-mvmul 20166 for the multiplication of a matrix with a column vector. In most cases, it is sufficient to regard members of ((Base‘𝑅) ↑𝑚 𝑁) as "column vectors", because ((Base‘𝑅) ↑𝑚 𝑁) is the base set of (𝑅 freeLMod 𝑁), see frlmbasmap 19922. See also the statements in [Lang] p. 508. | ||
Syntax | cmvmul 20165 | Syntax for the operator for the multiplication of a vector with a matrix. |
class maVecMul | ||
Definition | df-mvmul 20166* | The operator which multiplies an M x N -matrix with an N-dimensional vector. (Contributed by AV, 23-Feb-2019.) |
⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))) | ||
Theorem | mvmulfval 20167* | Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) ⇒ ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))) | ||
Theorem | mvmulval 20168* | Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) | ||
Theorem | mvmulfv 20169* | A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝐼 ∈ 𝑀) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) | ||
Theorem | mavmulval 20170* | Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) | ||
Theorem | mavmulfv 20171* | A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 18-Feb-2019.) (Revised by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) | ||
Theorem | mavmulcl 20172 | Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑𝑚 𝑁)) | ||
Theorem | 1mavmul 20173 | Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → ((1r‘𝐴) · 𝑌) = 𝑌) | ||
Theorem | mavmulass 20174 | Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴)) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌))) | ||
Theorem | mavmuldm 20175 | The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (𝐵 ↑𝑚 (𝑀 × 𝑁)) & ⊢ 𝐷 = (𝐵 ↑𝑚 𝑁) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → dom · = (𝐶 × 𝐷)) | ||
Theorem | mavmulsolcl 20176 | Every solution of the equation 𝐴∗𝑋 = 𝑌 for a matrix 𝐴 and a vector 𝐵 is a vector. (Contributed by AV, 27-Feb-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (𝐵 ↑𝑚 (𝑀 × 𝑁)) & ⊢ 𝐷 = (𝐵 ↑𝑚 𝑁) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐸 = (𝐵 ↑𝑚 𝑀) ⇒ ⊢ (((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑀 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐸)) → ((𝐴 · 𝑋) = 𝑌 → 𝑋 ∈ 𝐷)) | ||
Theorem | mavmul0 20177 | Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.) |
⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) | ||
Theorem | mavmul0g 20178 | The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.) |
⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑋 · 𝑌) = ∅) | ||
Theorem | mvmumamul1 20179* | The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.) |
⊢ × = (𝑅 maMul 〈𝑀, 𝑁, {∅}〉) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑𝑚 (𝑁 × {∅}))) ⇒ ⊢ (𝜑 → (∀𝑗 ∈ 𝑁 (𝑌‘𝑗) = (𝑗𝑍∅) → ∀𝑖 ∈ 𝑀 ((𝐴 · 𝑌)‘𝑖) = (𝑖(𝐴 × 𝑍)∅))) | ||
Theorem | mavmumamul1 20180* | The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maMul 〈𝑁, 𝑁, {∅}〉) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑𝑚 (𝑁 × {∅}))) ⇒ ⊢ (𝜑 → (∀𝑗 ∈ 𝑁 (𝑌‘𝑗) = (𝑗𝑍∅) → ∀𝑖 ∈ 𝑁 ((𝑋 · 𝑌)‘𝑖) = (𝑖(𝑋 × 𝑍)∅))) | ||
Syntax | cmarrep 20181 | Syntax for the row replacing function for a square matrix. |
class matRRep | ||
Syntax | cmatrepV 20182 | Syntax for the function replacing a column of a matrix by a vector. |
class matRepV | ||
Definition | df-marrep 20183* | Define the matrices whose k-th row is replaced by 0's and an arbitrary element of the underlying ring at the l-th column. (Contributed by AV, 12-Feb-2019.) |
⊢ matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)))))) | ||
Definition | df-marepv 20184* | Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | ||
Theorem | marrepfval 20185* | First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) | ||
Theorem | marrepval0 20186* | Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) | ||
Theorem | marrepval 20187* | Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))) | ||
Theorem | marrepeval 20188 | An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽))) | ||
Theorem | marrepcl 20189 | Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵) | ||
Theorem | marepvfval 20190* | First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) | ||
Theorem | marepvval0 20191* | Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))) | ||
Theorem | marepvval 20192* | Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) | ||
Theorem | marepveval 20193 | An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) | ||
Theorem | marepvcl 20194 | Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) | ||
Theorem | ma1repvcl 20195 | Closure of the column replacement function for identity matrices. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) | ||
Theorem | ma1repveval 20196 | An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), if(𝐽 = 𝐼, (1r‘𝑅), 0 ))) | ||
Theorem | mulmarep1el 20197 | Element by element multiplication of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → ((𝐼𝑋𝐿)(.r‘𝑅)(𝐿𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝐿)(.r‘𝑅)(𝐶‘𝐿)), if(𝐽 = 𝐿, (𝐼𝑋𝐿), 0 ))) | ||
Theorem | mulmarep1gsum1 20198* | The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽)) | ||
Theorem | mulmarep1gsum2 20199* | The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) | ||
Theorem | 1marepvmarrepid 20200 | Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.) |
⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) & ⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = 𝑋) |
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