Interpret matrices as rules that preserve linear combinations
A transformation T is linear when it preserves addition and scalar multiplication. Once bases are chosen, every finite-dimensional linear transformation has a matrix, and every matrix defines a linear transformation. The matrix columns record exactly where the basis vectors go.
The definition of linearity
Preserve combinations, not necessarily lengths or angles
Equivalently, T(c₁v₁+···+cₖvₖ)=c₁T(v₁)+···+cₖT(vₖ). Every linear map sends the zero vector to zero. A translation T(x)=Ax+b with b≠0 is affine, not linear.
Scaling
Diagonal matrices stretch or compress coordinate directions.
Rotation and reflection
Orthogonal matrices preserve lengths and angles.
Projection
A projection preserves one subspace and removes its orthogonal complement.
The standard matrix
Determine the entire map from its action on a basis
For T:Rⁿ→Rᵐ, the standard matrix is A=[T(e₁) T(e₂) … T(eₙ)]. Because every x is a linear combination of the standard basis vectors, T(x)=Ax.
If T(e₁)=(2,1) and T(e₂)=(−1,3), then A=[[2,−1],[1,3]]. For x=(x₁,x₂), T(x)=x₁T(e₁)+x₂T(e₂).
Kernel, image, and rank-nullity
Describe what disappears and what can be reached
Kernel N(T)
All input vectors mapped to zero. T is one-to-one exactly when N(T)={0}.
Image or range
All possible outputs T(x). For T(x)=Ax, the image is the column space C(A).
A map from Rⁿ to Rᵐ is onto when rank=m and one-to-one when rank=n. It can be both only when the relevant dimensions and rank permit it.
Interactive transformation laboratory
Watch a matrix move a grid, a unit square, and selected vectors
2D linear-transformation visualizer
The matrix is A=[[a,b],[c,d]]. Columns are the images of e₁ and e₂.
Composition and change of basis
Separate the abstract map from its coordinate description
Composition
If S acts first and T acts second, then [T∘S]=[T][S]. Matrix multiplication is built to represent this composition.
Change of basis
The same abstract transformation has different matrices in different bases. Similar matrices B=P⁻¹AP represent the same endomorphism under different coordinate systems.
Similarity preserves structural invariants
Similar matrices have the same characteristic polynomial, determinant, trace, eigenvalues, and rank, even though their entries may look very different.
Review questions
Test linearity, matrix representation, kernel, and image