Linear Transformations, Matrices and Change of Basis
Advanced linear algebra turns linear maps into structural invariants. Bases and matrices are coordinates; the underlying map or module is the object. Canonical forms are useful because they expose invariants that do not depend on a particular basis. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Learning pathAdvanced Linear Algebra
LevelAdvanced
FormatHandbook guide
Read time13 min
Executive summary
This chapter develops linear transformations, matrices and change of basis as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the field or coefficient ring and the vector space or module.
Choose bases only after deciding what structure should be preserved.
Represent the map by a matrix and track how the matrix changes under a basis change.
Use invariant subspaces, cyclic decomposition or elementary divisors to reduce the problem.
Read structural information from the resulting normal or canonical form.
Translate the matrix conclusion back into a basis-independent statement.
Core definitions
Definition
If U and W are subspaces of a vector space V , define U + W = {u + w : u ∈U and w ∈W}. 3.73 (i) Prove that U + W is a subspace of V . (ii) If U and U′ are subspaces of a finite-dimensional vector space V , prove that dim(U) + dim(U′) = dim(U ∩U′) + dim(U + U′). Hint. Take a basis of U ∩U′ and extend it to bases of U and of U′.
Definition
If U and W are vector spaces over a field k, then their direct sum is the set of all ordered pairs, U ⊕W = {(u, w) : u ∈U and w ∈W}, with addition (u, w) + (u′, w′) = (u + u′, w + w′) and scalar multiplication α(u, w) = (αu, αw). 3.74 If U and W are finite-dimensional vector spaces over a field k, prove that dim(U ⊕W) = dim(U) + dim(W). Linear Transformations Homomorphisms between vector spaces are called linear transformations.
Definition
If V is a vector space over a field k, then the general linear group, denoted by GL(V ), is the set of all nonsingular linear transformations V →V . A composite ST of linear transformations S and T is again a linear transformation, and ST is nonsingular if both S and T are; moreover, the inverse of a nonsingular linear transformation is again nonsingular. It follows that GL(V ) is a group with composition as operation, for composition of functions is always associative. We now show how to construct linear transformations T : V →W, where V and W are vector spaces over a field k. The next theorem says that there is a linear transformation that does anything to a basis.
Definition
Let X = v1, . . . , vn be a basis of V and let Y = w1, . . . , wm be a basis of W. If T : V →W is a linear transformation, then the matrix of T is the m × n matrix A = [ai j] whose jth column a1 j, a2 j, . . . , amj is the coordinate set of T (v j) determined by the w’s: T (v j) = m i=1 ai jwi. The matrix A does depend on the choice of bases X and Y; we will write A = Y [T ]X when it is necessary to display them. In case V = W, we often let the bases X = v1, . . . , vn and w1, . . . , wm coincide. Thus, I has 1’s on the diagonal and 0’s elsewhere. On the other hand, if X and Y are different bases, then Y [1V ]X is not the identity matrix; its columns are the coordinate sets of the x’s with respect to the basis Y.
Definition
Two n × n matrices B and A with entries in a field k are similar if there is a nonsingular matrix P with entries in k with B = P AP−1.
Definition
If T : V →W is a linear transformation, then the kernel (or the null space) of T is ker T = {v ∈V : T (v) = 0}, and the image of T is im T = {w ∈W : w = T (v) for some v ∈V }. As in Proposition 3.94, an m × n matrix A with entries in a field k determines a linear transformation kn →km, namely, y ↦Ay, where y is an n × 1 column vector. The proof of the next proposition is routine.
Definition
A linear transformation T : V →V is a scalar transformation if there is c ∈k with T (v) = cv for all v ∈V ; that is, T = c1V . A scalar matrix is a matrix of the form cI, where c ∈k and I is the identity matrix. A scalar transformation T = c1V is nonsingular if and only if c ̸= 0 (its inverse is c−11V ).
Definition
The commutative ring R/I constructed in Theorem 3.110 is called the quotient ring14 of R modulo I (briefly, R mod I). 14Presumably, quotient rings are so called in analogy with quotient groups.
Principal results and structural facts
Key result
Let U be a subspace of a vector space V of dimension n. (i) U is finite-dimensional and dim(U) ≤dim(V ). (ii) If dim(U) = dim(V ), then U = V .
Key result
Let v1, . . . , vn be a basis of a vector space V over a field k. If W is a vector space over k and u1, . . . , un is a list in W, then there exists a unique linear transformation T : V →W with T (vi) = ui for all i.
Key result
establishes the connection between linear transformations and matrices, and the definition of matrix multiplication arises from applying this construction to the composite of two linear transformations.
Key result
Let V and W be vector spaces over a field k, and let X = v1, . . . , vn and Y = w1, . . . , wm be bases of V and W, respectively. If Homk(V, W) denotes the set of all linear transformations T : V →W, and Matm×n(k) denotes the set of all m × n matrices with entries in k, then the function T ↦Y [T ]X is a bijection Homk(V, W) → Matm×n(k). Linear Algebra
Key result
Let T : V →W be a linear transformation of vector spaces V over a field k, and let X and Y be bases of V and W, respectively. If T is nonsingular, then the matrix of T −1 is the inverse of the matrix of T : X[T −1]Y = (Y [T ]X)−1.
Key result
Let T : V →V be a linear transformation on a vector space V over a field k. If X and Y are bases of V , then there is a nonsingular matrix P with entries in k so that Y [T ]Y = P ( X[T ]X ) P−1. Conversely, if B = P AP−1, where B, A, and P are n × n matrices with entries in k and P is nonsingular, then there is a linear transformation T : kn →kn and bases X and Y of kn such that B = Y [T ]Y and A = X[T ]X.
Key result
Let T : V →W be a linear transformation. (i) ker T is a subspace of V and im T is a subspace of W. (ii) T is injective if and only if ker T = {0}. We can now interpret the fact that a homogeneous system over a field k with r equations in n unknowns has a nontrivial solution if r < n. If A is the r × n coefficient matrix of the system, then ϕ : x ↦Ax is a linear transformation ϕ : kn →kr. If there is only the trivial solution, then ker ϕ = {0}, so that kn is isomorphic to a subspace of kr, contradicting
Key result
Let T : V →W be a linear transformation. (i) If T is nonsingular, then for every basis X = v1, v2, . . . , vn of V , we have T (X) = T (v1), T (v2), . . . , T (vn) a basis of W. (ii) Conversely, if there exists some basis X = v1, v2, . . . , vn of V for which T (X) = T (v1), T (v2), . . . , T (vn) is a basis of W, then T is nonsingular.
Key result
says that there is a linear transformation T : V →kn with T (vi) = ei for all i; by Lemma 3.103, T is nonsingular. •
Key result
does more than say that every finite-dimensional vector space is essentially the familiar vector space of all n-tuples. It says that a choice of basis in V is tantamount to a choice of coordinate set for each vector in V . We want the freedom to change coordinates because the usual coordinates may not be the most convenient ones for a given problem, as the reader has probably seen (in a calculus course) when rotating axes to simplify the equation of a conic section.
Key result
Let V be a finite-dimensional vector space with dim(V ) = n, and let T : V →V be a linear transformation. The following statements are equivalent: (i) T is an isomorphism; (ii) T is surjective; (iii) T is injective.
Key result
If A and B are n × n matrices with AB = I, then B A = I. Therefore, A is nonsingular with inverse B.
Key result
(i) The center of the group GL(V ) consists of all the nonsingular scalar transformations. (ii) The center of the group GL(n, k) consists of all the nonsingular scalar matrices.
Key result
If I is an ideal in a commutative ring R, then the additive abelian group R/I can be made into a commutative ring in such a way that the natural map π : R →R/I is a surjective ring homomorphism.
Source-grounded examples
Worked source example
(i) The identity function 1V : V →V on any vector space V is a nonsingular linear transformation. (ii) If θ is an angle, then rotation about the origin by θ is a linear transformation Rθ : R2 → R2. The function Rθ preserves addition because it takes parallelograms to parallelograms, and it preserves scalar multiplication because it preserves the lengths of arrows. (iii) If V and W are vector spaces over a field k, write Homk(V, W) for the set of all linear transformations V →W. Define addition S + T by v ↦S(v) + T (v) for all v ∈V , and define scalar multiplication αT : V →W, where α ∈k, by v ↦αT (v) for all v ∈V . Both S + T and αT are linear transformations, and Homk(V, W) is a vector space over k. ◀
Worked source example
, and hence it is a basis of V [because dim(V ) = 2]. Write v1 = v and v2 = T v. We compute X[T ]X. T (v1) = v2 and T (v2) = av1 + bv2 for some a, b ∈k. We conclude that X[T ]X = a b . ◀ The following proposition is a paraphrase of Theorem 3.92.
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
Failure mode
Control
Confusing a linear map with one particular matrix representing it.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Changing basis on only the domain or codomain when similarity requires a coordinated change.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Assuming diagonalisation when the polynomial or field conditions do not permit it.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Ignoring characteristic-dependent behaviour.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Treating canonical-form calculations as mere row reduction without tracking the allowed equivalence relation.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Verification checklist
The ambient set, ring, field, group, module or category has been stated.
Every operation and map used is well-defined in that setting.
The hypotheses of each structural result have been checked before use.
Representatives, coordinates or generators have not been confused with the underlying object.
Existence and uniqueness have been separated where both matter.
The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about linear transformations, matrices and change of basis?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.
Connections within the handbook
Source basis: supplied advanced algebra reference. Source-identifying authorship, publisher information, acknowledgements and biographical material are intentionally omitted. Mathematical terminology and results are retained in handbook form.
