Source: Friedberg, Insel & Spence, Linear Algebra 4th Ed.
Tags: invertible transformation, inverse matrix, isomorphism, isomorphic vector spaces, change of basis, change of coordinate matrix, similar matrices, matrix similarity, standard representation
Difficulty: Intermediate to Advanced Prerequisites: Sections 2.1–2.3 (linear transformations, null space, range, dimension theorem, matrix representation, composition, matrix multiplication). Solid grasp of bases, dimension, and the rank-nullity theorem.
Sections 2.4 and 2.5 close Chapter 2 by answering two fundamental questions. First: when can a linear transformation be "undone"? The answer is invertibility, which links one-to-one, onto, and full rank into a single equivalent condition for finite-dimensional spaces. This leads to isomorphisms, the formal way of saying two vector spaces are structurally identical. Second: what happens to the matrix of a linear operator when you switch to a different basis? The answer is similarity, the relation B = Q⁻¹AQ, which will dominate eigenvalue theory in later chapters.
A linear transformation between equal-dimension spaces is invertible precisely when it is one-to-one (equivalently, onto, equivalently, full rank). Invertible linear maps are called isomorphisms, and two finite-dimensional spaces are isomorphic if and only if they have the same dimension. Changing the basis of a linear operator conjugates its matrix by the change-of-coordinate matrix: [T]_{β'} = Q⁻¹[T]_β Q. Matrices related this way are called similar.
Invertible transformation
A linear map T: V → W for which there exists a function U: W → V with TU = I_W and UT = I_V. The function U is unique and is written T⁻¹. Think of it as T having a perfect "undo" button.
Inverse of a matrix, A⁻¹
For an n × n matrix A, an n × n matrix B such that AB = BA = Iₙ. If such B exists, A is called invertible and B is unique. Only square matrices can be invertible.
Isomorphism
An invertible linear transformation T: V → W. It is a structure-preserving bijection between vector spaces.
Isomorphic vector spaces
Two vector spaces V and W are isomorphic if there exists an isomorphism between them. In simple terms, they have the same structure and differ only in the labels on their vectors.
Standard representation of V with respect to β, φ_β
The function φ_β: V → Fⁿ defined by φ_β(x) = [x]_β. It is an isomorphism from any n-dimensional space to Fⁿ.
Change of coordinate matrix, Q = [I_V]^β_{β'}
The n × n invertible matrix that converts β'-coordinates into β-coordinates: [v]β = Q[v]{β'}. Its jth column is [x'_j]_β, the coordinate vector of the jth vector of β' written in the basis β.
Similar matrices
Matrices A and B in M_{n×n}(F) are similar if there exists an invertible Q with B = Q⁻¹AQ. Similarity is an equivalence relation. In simple terms, A and B represent the same linear operator in different bases.
T is invertible if and only if T is both one-to-one and onto.
Theorem 2.17: If T is linear and invertible, then T⁻¹ is also linear. The proof uses surjectivity and injectivity of T to show T⁻¹ preserves linear combinations.
For T: V → W with dim(V) = dim(W) finite, the following are all equivalent:
T is invertible
T is one-to-one
T is onto
rank(T) = dim(V)
N(T) = {0}
Key facts about inverses:
(TU)⁻¹ = U⁻¹T⁻¹ (order reverses)
(T⁻¹)⁻¹ = T
A is invertible if and only if L_A is invertible (Corollary 2 of Theorem 2.18), and (L_A)⁻¹ = L_{A⁻¹}.
Theorem 2.18: T is invertible if and only if [T]^γ_β is invertible. Furthermore, [T⁻¹]^β_γ = ([T]^γ_β)⁻¹. This links the inverse of a transformation directly to the inverse of its matrix.
For invertible matrices:
(AB)⁻¹ = B⁻¹A⁻¹ (order reverses)
(Aᵗ)⁻¹ = (A⁻¹)ᵗ
If A is invertible and AB = O, then B = O
Theorem 2.19: Two finite-dimensional vector spaces (over the same field) are isomorphic if and only if they have the same dimension. Dimension is the complete invariant.
Corollary: Every n-dimensional vector space over F is isomorphic to Fⁿ.
Examples of isomorphic pairs:
F² and P₁(F) (both dimension 2)
P₃(R) and M₂ₓ₂(R) (both dimension 4)
F³ and P₃(F) are NOT isomorphic (dimensions 3 vs. 4)
Theorem 2.20: The map Φ: L(V,W) → M_{m×n}(F) defined by Φ(T) = [T]^γ_β is an isomorphism. So the space of linear transformations is isomorphic to the space of matrices.
Corollary: dim(L(V,W)) = dim(V) · dim(W).
Theorem 2.21: φ_β: V → Fⁿ defined by φ_β(x) = [x]_β is an isomorphism.
The key relationship (the "commutative diagram"): L_A φ_β = φ_γ T, where A = [T]^γ_β. This means you can study T by working with L_A on Fⁿ and Fᵐ instead.
Theorem 2.22: Let Q = [I_V]^β_{β'}. Then:
Q is invertible
[v]β = Q[v]{β'} for every v in V
Q converts coordinates from the new basis β' to the old basis β. Its jth column is the old-basis representation of the jth new-basis vector.
Q⁻¹ does the reverse: it converts β-coordinates into β'-coordinates.
Theorem 2.23: If T is a linear operator on V, β and β' are ordered bases, and Q is the change of coordinate matrix from β' to β, then:
[T]_{β'} = Q⁻¹[T]_β Q
This is the similarity relation. It says the matrix of T in the new basis is obtained by conjugating the old matrix by Q.
To recover [T]β from [T]{β'}: [T]β = Q[T]{β'} Q⁻¹.
Worked example (reflection about y = 2x):
Choose β' = {(1,2), (−2,1)}, vectors along and perpendicular to the line.
In this basis, the reflection is diagonal: [T]_{β'} = [[1,0],[0,−1]].
The change-of-coordinate matrix to the standard basis β is Q = [[1,−2],[2,1]].
Then [T]β = Q[T]{β'}Q⁻¹ = (1/5)[[−3,4],[4,3]].
So T(a,b) = (1/5)(−3a + 4b, 4a + 3b).
Similarity is reflexive (A = I⁻¹AI), symmetric (if B = Q⁻¹AQ then A = QBQ⁻¹), and transitive.
Similar matrices share many properties: trace, determinant (Chapter 4), eigenvalues (Chapter 5), rank.
Key fact: tr(A) = tr(B) whenever A and B are similar.
Invertibility condition (finite-dimensional, equal dimensions): T invertible ⟺ one-to-one ⟺ onto ⟺ rank(T) = dim(V) ⟺ N(T) = {0}
Inverse of a matrix product: (AB)⁻¹ = B⁻¹A⁻¹
Inverse of a transpose: (Aᵗ)⁻¹ = (A⁻¹)ᵗ
Change of coordinate formula: [v]β = Q[v]{β'} where Q = [I_V]^β_{β'}
Similarity (change of basis for an operator): [T]_{β'} = Q⁻¹[T]_β Q
Dimension criterion for isomorphism: V ≅ W ⟺ dim(V) = dim(W) (finite-dimensional, same field)
Change of basis is used in engineering whenever you rotate a coordinate system to simplify a problem. In structural mechanics, stress tensors are represented by matrices, and rotating the coordinate axes to align with the principal directions diagonalises the matrix, making the physics transparent. The similarity relation Q⁻¹AQ is the same operation. In signal processing, choosing the Fourier basis transforms convolution (hard) into multiplication (easy), which is a change-of-basis argument at heart.
"Any matrix has an inverse." Only square matrices can be invertible, and not all square matrices are. A 2×3 matrix, for instance, has no inverse at all.
"If AB = I, I still need to check BA = I separately." For square matrices, one-sided invertibility implies two-sided invertibility (a consequence of the dimension theorem, via Theorem 2.5 and the L_A machinery). So AB = Iₙ alone is enough.
"Isomorphic means equal." Isomorphic spaces have the same structure but may consist of entirely different objects. R² and P₁(R) are isomorphic, but one contains ordered pairs and the other contains polynomials.
"The change of coordinate matrix depends on the transformation T." It does not. Q depends only on the two bases β and β'. The same Q works for converting coordinates of any vector, and for conjugating the matrix of any operator on that space.
"Similar matrices are equal." They represent the same transformation in different bases. They share invariants (trace, determinant, eigenvalues, rank) but can look very different entry by entry.
⚠️ "Is T invertible?" questions are answered by checking any of the equivalent conditions: null space trivial, rank equals dimension, one-to-one, or onto. Use whichever is easiest to verify.
⚠️ "Are V and W isomorphic?" reduces to comparing dimensions (for finite-dimensional spaces over the same field).
⚠️ Computing a change of coordinate matrix Q: write each new basis vector as a linear combination of the old basis vectors. The coefficients form the columns of Q.
⚠️ The similarity formula [T]_{β'} = Q⁻¹[T]_β Q is heavily tested. Be careful about the direction: Q changes β'-coordinates into β-coordinates. Getting Q and Q⁻¹ backwards is a common error.
⚠️ Similar matrices have equal traces, equal determinants, and equal eigenvalues. This is useful for quick "can these be similar?" checks.
1. True or false: A 3×4 matrix can be invertible.
A: False. Only square matrices can have inverses.
2. Fill in the blank: V is isomorphic to W if and only if ______.
A: dim(V) = dim(W) (assuming both are finite-dimensional over the same field).
3. True or false: If AB = Iₙ for n × n matrices A and B, then BA = Iₙ.
A: True. For square matrices, a one-sided inverse is a two-sided inverse.
4. Fill in the blank: [T]_{β'} = ______ [T]_β ______.
A: Q⁻¹ [T]_β Q, where Q is the change of coordinate matrix from β' to β.
5. True or false: Every 5-dimensional real vector space is isomorphic to R⁵.
A: True.
Q: Let T: R² → R³ be defined by T(a₁, a₂) = (a₁ − 2a₂, a₂, 3a₁ + 4a₂). Is T invertible?
A: dim(R²) = 2 and dim(R³) = 3. Since the dimensions are unequal, T cannot be invertible (an invertible map requires dim(V) = dim(W)).
Q: Verify that the inverse of A = [[5,7],[2,3]] is B = [[3,−7],[−2,5]].
A: AB = [[5·3 + 7·(−2), 5·(−7) + 7·5], [2·3 + 3·(−2), 2·(−7) + 3·5]] = [[1,0],[0,1]] = I₂. Likewise BA = I₂. So B = A⁻¹.
Q: In R², let β = {(1,1),(1,−1)} and β' = {(2,4),(3,1)}. Find the change of coordinate matrix Q that changes β'-coordinates into β-coordinates.
A: Express each β' vector in β-coordinates. (2,4) = 3(1,1) + (−1)(1,−1), so [(2,4)]_β = (3,−1)ᵗ. (3,1) = 2(1,1) + 1(1,−1), so [(3,1)]_β = (2,1)ᵗ. Therefore Q = [[3,2],[−1,1]].
Q: Let T be the linear operator on R² with [T]β = [[3,1],[−1,3]], using the basis β = {(1,1),(1,−1)}. Compute [T]{β'} where β' = {(2,4),(3,1)}, given that Q = [[3,2],[−1,1]] changes β'-coordinates to β-coordinates.
A: Q⁻¹ = (1/5)[[1,−2],[1,3]]. Then [T]_{β'} = Q⁻¹[T]_β Q = (1/5)[[1,−2],[1,3]] · [[3,1],[−1,3]] · [[3,2],[−1,1]] = [[4,1],[−2,2]].
Q: Which of the following pairs are isomorphic? (a) F³ and P₃(F). (b) F⁴ and P₃(F). (c) M₂ₓ₂(R) and P₃(R).
A: (a) dim(F³) = 3, dim(P₃(F)) = 4. Not isomorphic. (b) dim(F⁴) = 4, dim(P₃(F)) = 4. Isomorphic. (c) dim(M₂ₓ₂(R)) = 4, dim(P₃(R)) = 4. Isomorphic.
Invertibility connects to Chapter 3, where row reduction provides a concrete algorithm for computing A⁻¹. The determinant (Chapter 4) gives a scalar test: A is invertible if and only if det(A) ≠ 0. The similarity relation B = Q⁻¹AQ is the organising idea of Chapters 5 through 7: diagonalisation asks whether a matrix is similar to a diagonal matrix, and the answer depends on eigenvalues and eigenvectors. The change-of-coordinate matrix Q whose columns are eigenvectors is precisely the Q that diagonalises [T]_β.
invertible matrix, nonsingular matrix, matrix inverse, inverse transformation, isomorphism, isomorphic vector spaces, change of basis, change of coordinate matrix, similar matrices, similarity, conjugation, standard representation, coordinate vector, diagonalisation preview, Friedberg Chapter 2, Sections 2.4 and 2.5