Interactive guide
A matrix is just where it sends two arrows.
Drag either colored arrowhead. Everything else follows.
column 1 = (a, c) column 2 = (b, d) unit square, transformed
⎡ 2.00 0.00 ⎤ ⎣ 0.00 0.50 ⎦
—
- det(M)
- 1.00
- Area scale
- 1.00×
Every straight line stays straight and every grid line stays evenly spaced, that's what "linear" means. Once you know where a matrix sends just two vectors, (1,0) and (0,1), you know where it sends every point in the plane: any point is just some mix of those two, so its image is that same mix of where they landed.
The columns are the whole matrix
Write M = [[a, b], [c, d]]. Feed it (1,0) and out comes (a, c), the first column, read straight off the matrix. Feed it (0,1) and out comes (b, d), the second column. That's not a special property of those two vectors, it's the definition of matrix multiplication: a matrix's columns are just a record of where the basis vectors went.
Drag the blue arrow above and you're directly editing column 1 (a and c). Drag the orange one and you're editing column 2 (b and d). There's no other hidden state, those four numbers are the entire matrix.
Reading the picture
-
1
The grid shows the whole transformation
Every line stays straight; parallel lines stay parallel. That rigidity is exactly what "linear" buys you.
-
2
The purple square's new area is |det(M)|
Started as 1×1 = 1. Shrink the square and |det| < 1; stretch it and |det| > 1.
-
3
A negative det means it's flipped
The square turns inside out, a mirror image, not a rotation, even though the area formula doesn't care about the sign.
-
4
det = 0 means it's not reversible
The square collapses to a line or a point, an entire dimension of information is gone, so no inverse matrix can bring it back.
Try the Shear and Scale (2, 0.5) presets above, those are the exact same two matrices used on the Eigenvalues & Eigenvectors page. Everything you're watching here, the grid warp, the square's new area, is the setup; that page asks the next question: is there any direction this particular warp leaves pointing the same way, just longer or shorter? Drag the vector there and find out.
The jargons
A 2×2 matrix, fully explained by four numbers and what they do to a grid.
A mapping of the plane that keeps grid lines straight, evenly spaced, and parallel, and sends the origin to itself.
One column of the matrix, read top to bottom. It's exactly where that column's basis vector, (1,0) or (0,1), gets sent.
ad − bc. Its absolute value is the transformed unit square's area; its sign says whether orientation flipped.
One with det(M) = 0. It squashes the plane down to a line or a point and has no inverse.
A transformation that slides one axis sideways in proportion to the other, turning the unit square into a slanted parallelogram of the same area.
Whether "clockwise" stays clockwise. A negative determinant reverses it, the visual signature of a reflection.