To recap: a linear transformation is also known as a function that maps vectors to other vectors, and matrices can be thought of as linear transformations and coordinates

Matrix operations should coordinate with operations on Linear Transformations

Matrix operations Operations on Linear Transformations
Scalar multiplication Scalar multiplication
Addition/subtraction Addition/substraction
Multiplication of Matricies Function composition
Vector multiplication Functions

A linear transformation that maps a vector space onto itself must be represented by a square matrix

In matrix multiplication, and linear transformations in general, the order in which you do things matters because you can get different results

Matrix operations

Matrix addition is well defined when A and B have the same dimensions. Addition can be defined as (A + B)ij = Aij + Bij

Matrix addition is well defined when the number of columns in B is equal to the number of rows in A

Examples

Addition (and subtraction): (1231)+(−1430)=(0661)Multiplication: (−12106−431)(62−75−1−32−1)=(6(−1)−7(2)−1(1)+2(0)2(−1)+5(2)−3(1)−1(0)6(6)−7(−4)−1(3)+2(1)2(6)+5(−4)−3(3)−1(1))=(−21563−18)\text{Addition (and subtraction): } \begin{pmatrix} 1 & 2\\ 3 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 4\\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 6\\ 6 & 1 \end{pmatrix} \\ \text{Multiplication: } \begin{pmatrix} -1 & 2 & 1 & 0\\ 6 & -4 & 3 & 1 \end{pmatrix} \begin{pmatrix} 6 & 2 \\ -7 & 5 \\ -1 & -3 \\ 2 & -1 \end{pmatrix} = \begin{pmatrix} 6(-1) - 7(2) - 1(1) + 2(0) & 2(-1) + 5(2) - 3(1) - 1(0) \\ 6(6) - 7(-4) - 1(3) + 2(1) & 2(6) + 5(-4) - 3(3) - 1(1) \end{pmatrix} = \begin{pmatrix} -21 & 5\\ 63 & -18 \end{pmatrix}

Properties of Matrix Arithmetic

  • A + B = B + A
  • A + (B + C) = (A + B) + C
  • 0 + A = A
  • A - A = 0 λ(A+B)=λA+λB(1...0⋮⋱⋮0...1)∗A=A\lambda(A + B) = \lambda A + \lambda B \\ \begin{pmatrix} 1 & ... & 0 \\ \vdots & \ddots & \vdots \\ 0 & ... & 1 \end{pmatrix} * A = A