1. Vectors & linear combinations
Two vectors $\mathbf{v}$ and $\mathbf{w}$ live in the plane — drag their tips. Tune the scalars $a,b$ to form the combination $a\mathbf{v}+b\mathbf{w}$. When $\mathbf{v}$ and $\mathbf{w}$ are independent their combinations span all of $\mathbb{R}^2$; when they line up the span collapses to a line.
Drag the blue and amber arrowheads.
2. Dot product, angle & projection
Drag $\mathbf{a}$ and $\mathbf{b}$. The dot product $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$ measures alignment; cosine similarity is that value normalised. The dashed segment is the projection $\dfrac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}\,\mathbf{b}$ of $\mathbf{a}$ onto $\mathbf{b}$.
Orthogonal vectors have $\mathbf{a}\cdot\mathbf{b}=0$.
3. A matrix is a transformation
The $2\times2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ sends every point $\mathbf{x}$ to $A\mathbf{x}$. Its columns are the images of the basis vectors $\mathbf{e}_1,\mathbf{e}_2$. Tune the entries (or drag the column tips) and watch the unit grid and the cat-shape deform.
Drag the blue/amber column arrowheads.
4. Determinant = signed area
The columns of $A$ span a parallelogram. Its signed area is exactly $\det A=ad-bc$. A negative determinant means the transformation flips orientation; a zero determinant collapses the area to a line — the matrix is singular and non-invertible.
Drag either column arrowhead.
5. Composition is matrix multiplication
Applying $A$ then $B$ to the plane is the single transformation $BA$ — matrix multiplication is composition of maps (and order matters: $AB\ne BA$ in general). Press play to watch the grid rotate-then-shear continuously.
6. Gaussian elimination → RREF
Step through forward elimination and back-substitution on a $3\times4$ augmented matrix $[A\,|\,\mathbf{b}]$. Each step is an elementary row operation; the pivots march down the diagonal until the system is in reduced row-echelon form and the solution can be read off.
7. The inverse undoes the map
If $\det A\ne0$ the equation $A\mathbf{x}=\mathbf{b}$ has the unique solution $\mathbf{x}=A^{-1}\mathbf{b}$. Drag the target $\mathbf{b}$ (amber); the solved pre-image $\mathbf{x}$ (blue) is shown, and you can verify $A\mathbf{x}$ lands back on $\mathbf{b}$.
Drag the amber target. Singular $A$ has no inverse.
8. Eigenvalues & eigenvectors
An eigenvector $\mathbf{v}$ satisfies $A\mathbf{v}=\lambda\mathbf{v}$ — it keeps its direction, only stretching by $\lambda$. Drag the probe vector around the circle; when it lines up with an eigen-direction (highlighted) its image $A\mathbf{v}$ stays parallel.
Complex eigenvalues = pure rotation, no real axis.
9. Change of basis
A point has different coordinates in different bases. Drag the basis vectors $\mathbf{b}_1,\mathbf{b}_2$ and a point $\mathbf{p}$. The same point's coordinates in the $\{\mathbf{b}_1,\mathbf{b}_2\}$ basis are $[\mathbf{p}]_B=B^{-1}\mathbf{p}$.
Drag the two basis arrows and the green point.
10. The transformation zoo
Classic linear maps, animated from the identity to the target so you can see the motion. Each is a $2\times2$ matrix acting on the unit grid and a reference shape.
11. Gram–Schmidt orthonormalization
Gram–Schmidt turns two independent vectors into an orthonormal basis. Keep $\mathbf{q}_1=\mathbf{a}/\lVert\mathbf{a}\rVert$, then subtract from $\mathbf{b}$ its projection onto $\mathbf{q}_1$ and normalise: $\mathbf{q}_2=\dfrac{\mathbf{b}-(\mathbf{b}\cdot\mathbf{q}_1)\mathbf{q}_1}{\lVert\cdots\rVert}$.
Drag the blue ($\mathbf{a}$) and amber ($\mathbf{b}$) arrows.