Class 02/01/21

Linear Solution

- x_{1} + 3 x_{2} + 6 x + 3 - 7 x_{4} = 7 2 x_{1} - 4 x_{2} - 10 x_{3} + 10 x_{4} = - 8 S o l u t i o n : x_{1} = 2 + 3 t_{1} - t_{2} x_{2} = 3 - t_{1} + 2 t_{2} x_{3} = t_{1} x_{4} = t_{2} w h ere t_{1} an d t_{2} a re f ree p a r am e t ers M a t r i x f or m : B = [- 1 2 3 - 4 6 - 10 - 7 10] ⎣ ⎡ x_{1} x_{2} x_{3} x_{4} ⎦ ⎤ = [7 - 8]

Product of Matrices

k \times m ma t r i x = ⎣ ⎡ a_{11} a_{21} a_{k 1} a_{12} a_{22} a_{k 2} a_{1 m} a_{2 m} a_{km} ⎦ ⎤

First index is the row, second index is the column

Let A be an m x n matrix and B be an n x r matrix then AB is an m x r matrix whose ij entry is the product of the i'th row of A times the j'th column of B

B = [251830] ⎣ ⎡ 1 - 1 2 212 0 - 2 3 ⎦ ⎤ = [7 - 3 1118 7 - 16] 2 * 1 + 1 * - 1 + 3 * 2 = 7 2 * 2 + 1 * 1 + 3 * 2 = 11 2 * 0 + 1 * - 2 + 3 * 3 = 7

Ex:

⎣ ⎡ 147258369 ⎦ ⎤ ⎣ ⎡ 010 ⎦ ⎤ = ⎣ ⎡ 258 ⎦ ⎤

or (identity matrix)

⎣ ⎡ 147258369 ⎦ ⎤ ⎣ ⎡ 100010001 ⎦ ⎤ = ⎣ ⎡ 147258369 ⎦ ⎤

Properties

In general: AB $\neq =$ BA

A (B + C) = A B + A C it is associative (A + B) C = A C + BC = A BC

The transpose $A^{T}$ of A is the matrix obtained by flipping the rows and columns of A

[142536]^{T} = ⎣ ⎡ 123456 ⎦ ⎤ A^{T} = A (A B)^{T} = B^{T} A^{T}

A is symmetric if $A^{T} = A$ , example::

⎣ ⎡ 123204345 ⎦ ⎤

Suppose $v_{1} an d v_{2}$

A v_{1} = [23], A v_{2} = [50] then A (3 v_{1} + 2 v_{2}) = 3 A v_{1} + 2 A v_{2} = 3 [23] + 2 [50] = [169]

Augmented matrix:

$A x = b$

[A ∣ b]