3 Linear Algebra Review

Matrices and Vectors

The dimension of the matrix is going to be written as the number of row times the number of columns in the matrix.

A vector turns out to be a special case of a matrix.

Matrix with just one column is what we call a vector.

$$\begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} + \begin{bmatrix} 4 & 0.5 \\ 2 & 5 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0.5 \\ 4 & 10 \\ 3 & 2 \end{bmatrix}$$

Scalar Multiplication

$$3 * \begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ 6 & 15 \\ 9 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} * 3$$

Matrix Vector Multiplication

$$\begin{bmatrix} 1 & 3 \\ 4 & 0 \\ 2 & 1 \end{bmatrix} * \begin{bmatrix} 1 \\ 5 \end{bmatrix} = \begin{bmatrix} 16 = 1 * 1 + 3 * 5 \\ 4 = 4 * 1 + 0 * 5 \\ 7 = 2 * 1 + 1 * 5 \end{bmatrix}$$

Matrix Matrix Multiplication

$$\begin{bmatrix} C0 & C1 \\ C2 & C3 \end{bmatrix} = \begin{bmatrix} A0 & A1 \\ A2 & A3 \end{bmatrix} * \begin{bmatrix} B0 & B1 \\ B2 & B3 \end{bmatrix}$$

$$C0 = A0 * B0 + A1 * B2$$
$$C1 = A0 * B1 + A1 * B3$$
$$C2 = A2 * B0 + A3 * B2$$
$$C3 = A2 * B1 + A3 * B3$$

Matrix Multiplication Properties

$$A * B \neq B * A$$
$$A * (B * C) = (A * B) * C$$

Identity matrix, has the property that it has ones along the diagonals, right, and so on and is zero everywhere else.

$$AA^{-1} = A^{-1}A = I$$
$$AI = IA = A$$

Inverse and Transpose

Inverse

$$A^{-1}$$

Transpose

$$A^{T}$$ $$\begin{vmatrix} a & b\\ c & d\\ e & f \end{vmatrix} ^{T} = \begin{vmatrix} a & c & e\\ b & d & f \end{vmatrix}$$