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Linear Algebra
Step 8 of 22

The matrix

Understand first: linear transformations

You were probably taught to multiply a matrix by a vector like this: go across a row, down the vector, multiply and add. Take

A=[2113],x=[21]A = \begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix}, \qquad x = \begin{bmatrix} 2 \ 1 \end{bmatrix}

Row one gives 2(2)+1(1)=52(2) + 1(1) = 5. Row two gives 1(2)+3(1)=51(2) + 3(1) = 5. So Ax=(5,5)Ax = (5, 5).

It works, and it feels like a rule somebody invented. It isn't. Watch what happens if you compute the exact same thing a different way.

Take the columns instead. The columns of AA are (2,1)(2, 1) and (1,3)(1, 3). Now use the entries of xx as amounts: take 2 of the first column, 1 of the second.

2[21]+1[13]=[42]+[13]=[55]2\begin{bmatrix} 2 \ 1 \end{bmatrix} + 1\begin{bmatrix} 1 \ 3 \end{bmatrix} = \begin{bmatrix} 4 \ 2 \end{bmatrix} + \begin{bmatrix} 1 \ 3 \end{bmatrix} = \begin{bmatrix} 5 \ 5 \end{bmatrix}

The same answer, and this time you can see why.

Where the columns came from

Remember what x=(2,1)x = (2, 1) actually means: it is the recipe 2ı^+1ȷ^2\hat{\imath} + 1\hat{\jmath}. And the transformation is linear, which is exactly the promise that scaling and adding survive the trip:

A(2ı^+1ȷ^)=2 A(ı^)+1 A(ȷ^)A(2\hat{\imath} + 1\hat{\jmath}) = 2,A(\hat{\imath}) + 1,A(\hat{\jmath})

So the output is built from A(ı^)A(\hat{\imath}) and A(ȷ^)A(\hat{\jmath}) — where the two basis vectors landed. Feed in ı^=(1,0)\hat{\imath} = (1,0) and the row rule returns (2,1)(2, 1): the first column. Feed in ȷ^=(0,1)\hat{\jmath} = (0,1) and you get (1,3)(1, 3): the second column.

A matrix is a list of where the basis vectors land. Column one is the new ı^\hat{\imath}, column two is the new ȷ^\hat{\jmath}.

Multiplying by a vector is then just following that vector's own recipe to the new ingredients. The row-times-column rule is not a definition to memorise; it is the bookkeeping that falls out of this.

Check yourself

What does [0−110]\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} do? Read the columns: ı^\hat{\imath} lands on (0,1)(0,1), straight up; ȷ^\hat{\jmath} lands on (−1,0)(-1,0), pointing left. Both arrows turned a quarter turn counter-clockwise. It is a 90 degree rotation, and you never had to multiply anything to know that.