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All my math notes, now in Markdown.

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Matrices

Conceptually

Addition and Subtraction

Matrices of the same size can be added and subtracted together, by adding or subtracting each and every corresponding element.

\[\begin{bmatrix} 2 & 3 & 8\\ 12 & 8 & 5\\ 2 & 9 & 6\\ \end{bmatrix} + \begin{bmatrix} 8 & 7 & 1\\ 10 & 19 & 3\\ 7 & 5 & 2\\ \end{bmatrix} = \begin{bmatrix} 10 & 10 & 9\\ 22 & 27 & 8\\ 9 & 14 & 8\\ \end{bmatrix}\] \[\begin{bmatrix} 2 & 3 & 8\\ 12 & 8 & 5\\ 2 & 9 & 6\\ \end{bmatrix} - \begin{bmatrix} 8 & 7 & 1\\ 10 & 19 & 3\\ 7 & 5 & 2\\ \end{bmatrix} = \begin{bmatrix} -6 & -4 & 7\\ 2 & -11 & 2\\ -5 & 4 & 4\\ \end{bmatrix}\]

Scalar Multiplication

A matrix can be scaled up or down, by multiplying every element of the matrix by a scalar.

\[5 \begin{bmatrix} 3 & 6\\ 7 & 9\\ 12 & 3\\ \end{bmatrix} = \begin{bmatrix} 15 & 30\\ 35 & 45\\ 60 & 15\\ \end{bmatrix}\]

Matrix Multiplication

Two matrices, of varying size can be multiplied together, by taking the dot product of each row of the first and each column of the second.

These matrices must be “

The process for this is to multiply each element in each row of the first matrix by each element in each column of the second, and add the results together.

\[Taking\ the\ Dot\ Product\] \[\begin{bmatrix} \color{red}{1} & \color{red}{2} & \color{red}{3}\\ 4 & 5 & 6\\ \end{bmatrix} \begin{bmatrix} \color{blue}{7} & 8\\ \color{blue}{9} & 10\\ \color{blue}{11} & 12\\ \end{bmatrix} = \begin{bmatrix} \color{purple}{58} & 64\\ 139 & 154\\ \end{bmatrix}\] \[(1)(7) + (2)(9) + (3)(11) = 58\]

The resulting matrix will have the height of the first matrix, and the width of the second.

Non-commutativity

Matrix multiplication is not commutative. $[A][B]$ may $= [C]$, but $[B][A]$ is not obligated to $=[C]$.

Distributivity

Although matrix multiplication is non-commutative, it does keep the distributative property.

As

What $[A][B]$ represents: do transformation $B$, then transformation $A$.

(AB)X = A(BX)