How a linear transformation changes the plane

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Math Workspace (CAS)

The CAS embed makes the rule concrete. Multiplying the matrix by $e_{1}$ , $e_{2}$ , and $e_{1} + e_{2}$ shows how the entire plane is rebuilt from two image vectors.

Two vectors control the picture

For a matrix $A = [2111]$ , the first column is $A e_{1}$ and the second column is $A e_{2}$ .

That means the standard horizontal unit vector moves to $[21]$ , while the vertical unit vector moves to $[11]$ .

What happens to the grid

Every vector $[x y]$ is $x e_{1} + y e_{2}$ . Because the transformation is linear, its image is $x A e_{1} + y A e_{2}$ .

So straight grid lines stay straight, parallel lines stay parallel, and the unit square becomes the parallelogram spanned by the two column vectors.

Area and orientation

For this matrix, $det (A) = 1$ . The shape of the grid changes, but signed area is preserved.

A different matrix could stretch area, shrink it, flip orientation, or collapse the plane onto a line. The determinant records that global area behavior.

Common Pitfall

A linear transformation does not have to preserve angles or lengths. It preserves linear combinations and sends the origin to the origin; rotations and reflections are only special cases.

Try a Variation

Change the matrix to $[1021]$ . Where do $e_{1}$ and $e_{2}$ go, and what happens to the unit square?

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