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Back to Linear Algebra

Matrix as a Transformation

A matrix is not a table of numbers to multiply. It is a way of moving the whole plane — and its two columns say exactly where î and ĵ go.

Step 1 of 22

Speed

This grid is the plane before anything happens. î = (1, 0) is green and ĵ = (0, 1) is red. Every vector is a mix of the two: v = (2, 1) means 2 steps of î plus 1 step of ĵ.

M =

det M = 1the shaded square's area, signed

\begin{bmatrix} \textcolor{#4ade80}{1} & \textcolor{#f87171}{0} \\ \textcolor{#4ade80}{0} & \textcolor{#f87171}{1} \end{bmatrix}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = 2\textcolor{#4ade80}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}} + 1\textcolor{#f87171}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}

The colours carry the idea: the green column is where î lands, the red column is where ĵ lands, and Mv is just v's recipe — 2 of the green plus 1 of the red.

Matrix as a Transformation

A matrix is not a table of numbers to multiply. It is a way of moving the whole plane — and its two columns say exactly where î and ĵ go.

Step 1 of 22

Speed

This grid is the plane before anything happens. î = (1, 0) is green and ĵ = (0, 1) is red. Every vector is a mix of the two: v = (2, 1) means 2 steps of î plus 1 step of ĵ.

M =

det M = 1the shaded square's area, signed

\begin{bmatrix} \textcolor{#4ade80}{1} & \textcolor{#f87171}{0} \\ \textcolor{#4ade80}{0} & \textcolor{#f87171}{1} \end{bmatrix}\begin{bmatrix} 2 \\ 1 \end{bmatrix} = 2\textcolor{#4ade80}{\begin{bmatrix} 1 \\ 0 \end{bmatrix}} + 1\textcolor{#f87171}{\begin{bmatrix} 0 \\ 1 \end{bmatrix}} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}

The colours carry the idea: the green column is where î lands, the red column is where ĵ lands, and Mv is just v's recipe — 2 of the green plus 1 of the red.

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Matrix as a Transformation

See also

Matrix as a Transformation

See also