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I am familiar using equations like $$y = \sum_i a_ix_i$$ such as $$y = mx + b$$ I am also familiar with multiplying matrices of the form

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} B= \begin{bmatrix} b_{1} \\ b_{2} \\ b_{3} \\ \end{bmatrix}$$ such that $$AB = \begin{bmatrix} a_{11}b_{1}+ a_{12}b_{2}+ a_{13}b_{3} \\ a_{21}b_{1}+ a_{22}b_{2}+ a_{23}b_{3} \\ a_{31}b_{1}+ a_{32}b_{2}+ a_{33}b_{3} \\ \end{bmatrix}$$

However, I am confused how to relate geometrically my intuitions about linear equations to matrices.

When I envision a linear equation, I understand directly how adjusting the different parameters changes the line. Increasing $b$ moves the y-intercept up. changing $m$ causes the line to pivot. But I have trouble envisioning these geometrical relationships with matrices.

My Question

Is there an intuitive geometric way to understand the relationship between single linear equations and equations with matrices and vectors? How do I make sense of operations like rotation and translation in the context of matrices?

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Observe these three products of your matrix $A$ with the three basis vectors:

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \end{bmatrix} $$ $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \end{bmatrix} $$ $$ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{13} \\ a_{23} \\ a_{33} \end{bmatrix} $$

So the columns of $A$ are simply the answer to the question, "What does this matrix do to the three basis vectors?"

If you think of the three basis vectors as three edges of a cube, then multiplication by the matrix will rotate and/or deform the cube as it rotates the three basis vectors, changes their lengths, and/or alters the angles between them.

The kinds of transformations you can get are much more varied than what you can do simply by changing the $m$ in $y=mx+b$. The best way to understand the geometric meaning of matrices may be to work out the matrices for various geometric transformations: rotations around each axis, shears parallel to planes, etc. It will still be difficult to look at an arbitrary matrix and see exactly what it does, partly because the kinds of transformations you're able to do are so much richer than just changing the slope of a line, and the descriptions of them can be that much more complicated.

Translations are less frequently done with matrices. You can use a matrix notation for translations, but it usually involves adding an extra entry in your vectors (which always has the value $1$) and and extra column and row in your matrices. For example, the effect on the first basis vector would be:

$$ \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{11} + a_{14} \\ a_{21} + a_{24} \\ a_{31} + a_{34} \\ 1 \end{bmatrix} $$ That is, the last column represents the amount by which the result is translated, in addition to whatever other effects the matrix has.

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