Matrix as a Linear Transformation
A 2x2 matrix bends the whole plane while keeping every line straight. Drag where the basis vectors land and watch the grid follow, the determinant as signed area, and the eigenvectors as the directions that refuse to turn.
Matrix as a Linear Transformation
A 2x2 matrix is not a static grid of numbers, it is an instruction for moving every point of the plane at once. The rule keeps lines straight, keeps parallel lines parallel, and leaves the origin where it is. Drag the labelled tips of the two basis vectors, or type the entries, and read the geometry off the numbers as the grid and a sample shape bend with them.
Rotation has complex eigenvalues and determinant one. Pure scaling stretches the two axes by different factors. Shear slides one axis while the other holds still. Reflection flips orientation, so its determinant is negative. The singular preset collapses the whole plane onto a line, so its determinant is zero.
An eigenvector is a direction the transformation does not turn, it only stretches or squashes along that line, and the eigenvalue is the factor. On the plane the dashed green line is the invariant direction and the solid green stub is the image of the unit eigenvector, so its length is the eigenvalue. When the eigenvalue is negative the stub points the opposite way, because the direction is preserved but flipped end to end.
The shaded parallelogram is the image of the unit square whose sides are i-hat and j-hat. Its area is the absolute value of the determinant, 1.00 here, and that is exactly how much the transformation multiplies every area in the plane. When the determinant turns negative the fill switches colour, which marks the moment the plane gets flipped over and orientation reverses.
Right now the determinant is positive, so orientation is preserved. Drag a tip until the i and j vectors line up and the determinant passes through zero, the instant the plane collapses to a line.
Every grid line, the sample shape, and both basis vectors are transformed by the same matrix-vector product. Determinant is ad − bc; eigenpairs come from solving λ² − (trace)λ + det = 0 in closed form.