Source: Friedberg, Insel & Spence, Linear Algebra 4th Ed.
Tags: matrix representation, ordered basis, coordinate vector, matrix multiplication, composition of linear transformations, left-multiplication transformation, identity matrix, Kronecker delta, incidence matrix
Difficulty: Intermediate Prerequisites: Section 2.1 (linear transformations, null space, range, dimension theorem). Familiarity with bases, dimension, and span from Chapter 1.
Section 2.1 studied linear transformations abstractly, through their null spaces and ranges. Sections 2.2 and 2.3 make everything computational by showing how to represent any linear transformation as a matrix once you choose ordered bases. The payoff is enormous: abstract questions about transformations become concrete questions about matrices, and vice versa. Section 2.3 then shows that composing two linear transformations corresponds to multiplying their matrices, which is precisely why matrix multiplication is defined the way it is.
Once you fix ordered bases for the domain and codomain, every linear transformation corresponds to a unique matrix whose columns are the coordinate vectors of the images of the basis vectors. Adding transformations corresponds to adding matrices; composing transformations corresponds to multiplying matrices. The left-multiplication transformation L_A connects any matrix A to a concrete linear map, letting you transfer results between the two settings freely.
Ordered basis
A basis for V together with a specific ordering of its vectors. The order matters: {e₁, e₂, e₃} and {e₂, e₁, e₃} are different ordered bases even though they contain the same vectors. Think of it as a numbered list rather than a set.
Standard ordered basis
For Fⁿ, the ordered basis {e₁, e₂, …, eₙ}. For Pₙ(F), the ordered basis {1, x, x², …, xⁿ}.
Coordinate vector of x relative to β, [x]_β
If β = {u₁, …, uₙ} is an ordered basis for V and x = a₁u₁ + … + aₙuₙ, then [x]_β is the column vector (a₁, a₂, …, aₙ)ᵗ. In simple terms, it is the list of coefficients when you write x as a linear combination of the basis vectors, stacked into a column.
Matrix representation of T in the ordered bases β and γ, [T]^γ_β
The m × n matrix A whose jth column is [T(v_j)]_γ, where β = {v₁, …, vₙ} is the ordered basis for V and γ = {w₁, …, wₘ} is the ordered basis for W. When V = W and β = γ, we write [T]_β. This matrix encodes everything T does.
L(V, W)
The vector space of all linear transformations from V into W. When V = W, we write L(V).
Composition of linear transformations, UT
The composite function (UT)(x) = U(T(x)). If T: V → W and U: W → Z are both linear, then UT: V → Z is also linear (Theorem 2.9).
Matrix multiplication (product AB)
For A an m × n matrix and B an n × p matrix, the product AB is the m × p matrix with (AB){ij} = sum from k=1 to n of A{ik} B_{kj}. The (i,j) entry is the dot product of row i of A with column j of B.
Kronecker delta, δ_{ij}
Equals 1 when i = j and 0 when i ≠ j. Used to define the identity matrix concisely.
Identity matrix, Iₙ
The n × n matrix with (Iₙ){ij} = δ{ij}. Ones on the diagonal, zeros elsewhere. It acts as the multiplicative identity for square matrices.
Left-multiplication transformation, L_A
For an m × n matrix A, the function L_A: Fⁿ → Fᵐ defined by L_A(x) = Ax. This is the bridge between matrices and linear transformations on Fⁿ.
Choose ordered bases β = {v₁, …, vₙ} for V and γ = {w₁, …, wₘ} for W.
Compute T(v_j) for each basis vector v_j and write it as a linear combination of the w_i.
The coefficients form the jth column of [T]^γ_β.
Changing the ordered bases changes the matrix. The same transformation can have different matrix representations.
Worked example (differentiation): T: P₃(R) → P₂(R) defined by T(f(x)) = f'(x), with standard ordered bases β and γ.
T(1) = 0 = 0·1 + 0·x + 0·x²
T(x) = 1 = 1·1 + 0·x + 0·x²
T(x²) = 2x = 0·1 + 2·x + 0·x²
T(x³) = 3x² = 0·1 + 0·x + 3·x²
Result: [T]^γ_β is the 3×4 matrix with rows [0,1,0,0], [0,0,2,0], [0,0,0,3].
Theorem 2.8: [T + U]^γ_β = [T]^γ_β + [U]^γ_β and [aT]^γ_β = a[T]^γ_β. So the map T → [T]^γ_β is itself a linear transformation from L(V,W) into M_{m×n}(F).
Theorem 2.9: The composition UT of linear maps T: V → W and U: W → Z is linear.
Theorem 2.10: Composition distributes over addition, is associative, and the identity acts as an identity. These are the same algebraic properties that matrix multiplication inherits.
The definition of (AB){ij} = sum of A{ik} B_{kj} is chosen precisely so that [UT]^γ_α = [U]^γ_β · [T]^β_α (Theorem 2.11).
Matrix multiplication encodes function composition. This is the reason the rule exists, not an arbitrary convention.
Not commutative: AB ≠ BA in general, even when both products are defined.
Associative: A(BC) = (AB)C (Theorem 2.16, proved elegantly via left-multiplication transformations).
Distributive: A(B + C) = AB + AC and (D + E)A = DA + EA.
Identity: I_m A = A = A I_n.
Transpose of a product: (AB)ᵗ = BᵗAᵗ. The order reverses.
No cancellation law: A² = O does not imply A = O.
Theorem 2.14: [T(u)]_γ = [T]^γ_β · [u]_β. To find the coordinate vector of T(u) in the codomain basis, multiply the matrix representation by the coordinate vector of u in the domain basis.
Theorem 2.15 (key properties):
[L_A]^γ_β = A (the matrix representation of L_A with respect to the standard bases is just A itself)
L_A = L_B if and only if A = B
L_{A+B} = L_A + L_B and L_{aA} = aL_A
Every linear map T: Fⁿ → Fᵐ equals L_C for a unique matrix C = [T]^γ_β
L_{AE} = L_A L_E (composition corresponds to multiplication)
L_{Iₙ} = I_{Fⁿ}
This tool lets you prove matrix theorems by translating them into statements about linear transformations and using the algebraic properties of composition. The associativity of matrix multiplication (Theorem 2.16) is proved this way.
Theorem 2.13: Column j of AB equals A times column j of B. Row i of AB is a linear combination of the rows of B with coefficients from row i of A.
Matrix representation (jth column): jth column of [T]^γ_β = [T(v_j)]_γ
Evaluating T via its matrix: [T(u)]_γ = [T]^γ_β · [u]_β
Composition as matrix product: [UT]^γ_α = [U]^γ_β · [T]^β_α
Matrix multiplication entry formula: (AB){ij} = sum from k=1 to n of A{ik} B_{kj}
Transpose of a product: (AB)ᵗ = BᵗAᵗ
Size mnemonic: (m × n) · (n × p) = (m × p); the inner dimensions must match.
Matrix multiplication appears in virtually every computational field. When a graphics pipeline transforms a 3D object, it multiplies coordinate vectors by a sequence of transformation matrices (rotation, scaling, translation). Composing several transformations into one matrix (the model-view-projection matrix) is more efficient than applying them separately, and this works precisely because composition equals matrix multiplication. In machine learning, every layer of a neural network applies a left-multiplication transformation followed by a nonlinearity.
"The matrix representation of T is unique." It depends on the choice of ordered bases. Change the bases and you change the matrix (Section 2.5 makes this precise).
"Matrix multiplication is commutative." It is not. AB and BA can be entirely different, or one may not even be defined when the other is.
"If AB = O, then A = O or B = O." False. Cancellation does not hold for matrices. There exist nonzero matrices A and B with AB = O.
"The size of [T]^γ_β is dim(V) × dim(W)." The size is dim(W) × dim(V), i.e. m × n, not n × m. The number of rows comes from the codomain, the number of columns from the domain.
⚠️ "Compute [T]^γ_β" is a bread-and-butter exam question. Apply T to each domain basis vector, express the result in the codomain basis, and read off the columns.
⚠️ The relationship [UT]^γ_α = [U]^γ_β [T]^β_α is frequently tested. Pay attention to which bases go where; the "inner" basis (β) must match.
⚠️ Know Theorem 2.14: [T(u)]_γ = [T]^γ_β [u]_β. This is how you use a matrix to compute what T does to a specific vector.
⚠️ Matrix multiplication is associative but not commutative. Exam true/false questions love this distinction.
⚠️ (AB)ᵗ = BᵗAᵗ (order reverses). This is tested frequently and often confused.
1. True or false: If m = dim(V) and n = dim(W), then [T]^γ_β is an m × n matrix.
A: False. It is n × m (dim(W) rows, dim(V) columns). Actually, correction: [T]^γ_β has dim(W) = m rows and dim(V) = n columns, so it is m × n only if you define m = dim(W) and n = dim(V). The matrix has as many columns as the domain dimension and as many rows as the codomain dimension.
2. Fill in the blank: The jth column of [T]^γ_β is ______.
A: [T(v_j)]_γ, the coordinate vector of T(v_j) relative to γ.
3. True or false: AB = BA for all square matrices A and B.
A: False. Matrix multiplication is not commutative.
4. True or false: [T + U]^γ_β = [T]^γ_β + [U]^γ_β.
A: True (Theorem 2.8).
5. Fill in the blank: (AB)ᵗ = ______.
A: BᵗAᵗ
Q: Let T: R² → R³ be defined by T(a₁, a₂) = (a₁ + 3a₂, 0, 2a₁ − 4a₂). Let β and γ be the standard ordered bases. Compute [T]^γ_β.
A: T(1,0) = (1, 0, 2) and T(0,1) = (3, 0, −4). So [T]^γ_β has columns (1, 0, 2)ᵗ and (3, 0, −4)ᵗ, giving the 3×2 matrix with rows [1, 3], [0, 0], [2, −4].
Q: Let T: P₃(R) → P₂(R) be differentiation. Use the matrix representation to compute T(2 − 4x + x² + 3x³).
A: With standard ordered bases, [T]^γ_β is the 3×4 matrix [0,1,0,0; 0,0,2,0; 0,0,0,3]. The coordinate vector of 2 − 4x + x² + 3x³ is (2, −4, 1, 3)ᵗ. Multiplying: (−4, 2, 9)ᵗ, which corresponds to −4 + 2x + 9x². This matches the derivative directly.
Q: Let A = [[1,2,1],[0,4,−1]] and B = [[4],[2],[5]]. Compute AB.
A: AB = [[1·4 + 2·2 + 1·5], [0·4 + 4·2 + (−1)·5]] = [[13], [3]].
Q: Find matrices A and B such that AB = O but A ≠ O and B ≠ O.
A: Take A = [[0,1],[0,0]] and B = [[0,0],[0,0]]... actually use A = [[0,1],[0,0]] and B = [[0,0],[0,0]]. Better example: A = [[1,1],[0,0]], B = [[1,−1],[−1,1]]. Then AB = [[0,0],[0,0]] = O, but neither A nor B is zero.
The matrix representation is the bridge between the abstract theory of Chapter 1 and 2.1 and the computational machinery of Chapter 3 (solving systems via row reduction). The left-multiplication transformation L_A reappears in Section 2.4 (invertibility) and is central to proving that matrix multiplication is associative. Section 2.5 explores what happens to the matrix representation when you change bases, leading to the concept of matrix similarity, which dominates Chapters 5, 6, and 7 (eigenvalues, diagonalisation, canonical forms).
matrix representation, ordered basis, coordinate vector, standard basis, matrix multiplication, composition, left-multiplication transformation, identity matrix, Kronecker delta, transpose of product, L_A, incidence matrix, dominance relation, clique, Friedberg Chapter 2, Sections 2.2 and 2.3