Linear Algebra Visualizer -- 2D Matrix Transformations Animated

Understanding Linear Transformations

What is a Matrix Transformation?

A matrix transformation is a function that maps every point in space to a new location using matrix multiplication. In 2D, a 2x2 matrix defines where the basis vectors (1,0) and (0,1) land, and all other vectors follow the grid.

Matrix Multiplication Visualized

When you multiply a matrix by a vector, you are asking: "Where does this vector land after the transformation?" The columns of the matrix tell you where the basis vectors go, and linear combinations do the rest.

The Determinant

The determinant tells you how much the transformation scales areas. A determinant of 2 means all areas double. A determinant of 0 means the space collapses to a line or point (the transformation is not invertible). A negative determinant means the transformation flips orientation (like turning a shape inside-out).

Eigenvectors and Eigenvalues

Eigenvectors are special directions that only get scaled, not rotated. The eigenvalue is the scaling factor. Visually, eigenvectors are the "axes of the transformation" -- they point in directions that remain unchanged except for stretching or compressing.

For a rotation (other than 180 degrees), there are no real eigenvectors because every direction rotates. For a scaling matrix, the eigenvectors are the coordinate axes themselves.

Common Transformation Matrices

[[cos(theta), -sin(theta)],
 [sin(theta),  cos(theta)]]

[[sx, 0 ],
 [0,  sy]]

[[1, k],
 [0, 1]]

[[1,  0],
 [0, -1]]

Composition of Transformations

You can chain multiple transformations by multiplying their matrices. The order matters: rotating then scaling is different from scaling then rotating. Matrix multiplication is read right-to-left: M2 * M1 means "apply M1 first, then M2".

Frequently Asked Questions

What is a matrix transformation?

A matrix transformation is a function that maps vectors from one space to another using matrix multiplication. In 2D, a 2x2 matrix can rotate, scale, shear, or reflect vectors. The transformation is applied by multiplying the matrix by each vector.

What does the determinant of a matrix tell you visually?

The determinant represents how much the matrix scales areas. A determinant of 2 doubles all areas; a determinant of 0.5 halves them; a determinant of 0 collapses everything to a line or point; a negative determinant means the transformation flips orientation.

What are eigenvectors?

Eigenvectors are vectors that only get scaled (not rotated) when a matrix transformation is applied. The eigenvalue is the scaling factor. Visually, eigenvectors point in directions that remain unchanged by the transformation -- they just stretch or shrink along those directions.

Does this tool store any data?

No. All visualizations run entirely in your browser. No data is sent to any server.

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