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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.

Addition and Scalar Multiplication

Addition

[latex] \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} [/latex]

Scalar Multiplication

[latex] 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 [/latex]

Matrix Vector Multiplication

[latex] \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} [/latex]

Matrix Matrix Multiplication

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

[latex]C0 = A0 * B0 + A1 * B2[/latex] [latex]C1 = A0 * B1 + A1 * B3[/latex] [latex]C2 = A2 * B0 + A3 * B2[/latex] [latex]C3 = A2 * B1 + A3 * B3[/latex]

Matrix Multiplication Properties

[latex]A * B \neq B * A[/latex] [latex]A * (B * C) = (A * B) * C[/latex] Identity matrix, has the property that it has ones along the diagonals, right, and so on and is zero everywhere else. [latex]AA^{-1} = A^{-1}A = I[/latex] [latex]AI = IA = A[/latex]

Inverse and Transpose

Inverse

[latex]A^{-1}[/latex]

Transpose

[latex]A^{T}[/latex]

[latex] \begin{vmatrix} a & b\\ c & d\\ e & f \end{vmatrix} ^{T} = \begin{vmatrix} a & c & e\\ b & d & f \end{vmatrix} [/latex]