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Linear Algebra Unit 5: Linear transformations

A linear transformation preserves addition and scaling. Every one between finite-dimensional spaces is a matrix.

Covers: Matrix of a transformation, kernel & image, change of basis

Study guide

A linear transformation T satisfies T(u + v) = T(u) + T(v) and T(cu) = cT(u). Consequences: T(0) = 0, and T is determined by what it does to a basis. Transformations like T(x) = x + 1 that move the origin aren't linear.

Every linear transformation from ℝⁿ to ℝᵐ is multiplication by a matrix, its standard matrix, whose columns are T(e₁), T(e₂), …, T(eₙ), the images of the standard basis vectors. This turns geometric transformations into matrices: rotations, reflections, shears, projections and scalings. Composition of transformations corresponds to multiplying their matrices, applied right to left.

The kernel (null space) of T is the set of vectors sent to 0, and the image (range) is the set of all outputs, the column space of the matrix. T is one-to-one exactly when its kernel is only {0}, equivalently the matrix's columns are linearly independent. T is onto exactly when its image is all of ℝᵐ, equivalently the columns span ℝᵐ (a pivot in every row).

A change of basis describes vectors with respect to a different basis B. The change-of-coordinates matrix P has the B vectors as columns, so x = P[x]_B. A transformation's matrix in basis B is P⁻¹AP. Choosing a good basis, such as one made of eigenvectors, can make a transformation's matrix much simpler.

A linear transformation T: ℝⁿ → ℝⁿ is invertible exactly when it is one-to-one and onto, which happens exactly when its standard matrix A is invertible. The inverse transformation has standard matrix A⁻¹. A map between spaces of different dimensions can never be invertible. Rotation by θ counterclockwise has matrix [cos θ −sin θ; sin θ cos θ], and its inverse is rotation by −θ, which equals its transpose. Reflection across the x-axis is [1 0; 0 −1], and projection onto the x-axis is [1 0; 0 0].

A matrix of the form [a −b; b a] equals r times a rotation, where r = √(a² + b²) and the angle θ satisfies cos θ = a/r and sin θ = b/r. Its eigenvalues are the complex numbers a ± bi. A and B are similar if B = P⁻¹AP for some invertible P. In that case they represent the same transformation in different bases, so they have the same characteristic polynomial, eigenvalues, determinant, trace and rank.

The composition S∘T (first T, then S) is defined only when T's codomain is S's domain. If T: ℝⁿ → ℝᵐ has matrix A and S: ℝᵐ → ℝᵖ has matrix B, then S∘T: ℝⁿ → ℝᵖ has the p×n standard matrix BA.

Key ideas

• T is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u).
• The standard matrix has columns T(e₁), T(e₂), ….
• Kernel: vectors sent to 0. Range: all outputs.
• T is one-to-one ⇔ kernel is {0}; onto ⇔ range is the whole codomain.
• Change of basis rewrites coordinates: [x]_B = P_B⁻¹x.

Formulas and rules

Common mistakes

• Calling T(x) = x + b linear. A shift by b ≠ 0 isn't linear.
• Mixing up the kernel and the range.

Words to know

Two conditions for linearity

T(u + v) = T(u) + T(v) and T(cu) = cT(u).

Columns of the standard matrix

T(e₁), T(e₂), …, T(eₙ)

Kernel of T

All vectors x with T(x) = 0.

One-to-one transformation

Kernel is only the zero vector.

Linear transformation

T(u + v) = T(u) + T(v), T(cu) = cT(u); must send 0 to 0.

Standard matrix

The matrix with columns T(e₁), …, T(eₙ).

Image (range) of T

All outputs T(x): the column space of its matrix.

Onto transformation

Every vector in the codomain is an output (a pivot in every row).

Composition of transformations

Multiply their matrices, applied right to left.

Coordinate vector [x]_B

The weights that build x from the basis B.

Change-of-coordinates matrix

P with the B vectors as columns: x = P[x]_B.

Matrix of T relative to B

P⁻¹AP

Sample practice questions

27 more questions like these, from level 1 to 5, are in the practice set.