Problem 1 — Injective, Surjective, Bijective from a Matrix
A map is injective if forces (distinct inputs give distinct outputs), surjective if (every target vector is hit), and bijective if it is both.
Let and have matrices
(a) For each map, compute and a basis of , then decide whether it is injective, surjective, or bijective.
(b) One of these maps can never be injective and the other can never be surjective, no matter which matrix you chose. Using rank-nullity, explain why a linear map is never injective and a linear map is never surjective.
(c) Prove: if is linear and injective, and is linearly independent in , then is linearly independent in .
Problem 2 — The Inverse of a Linear Map
A linear map is invertible if there is a map with and . Such an is unique, written , and undoes . (Note has nothing to do with .)
Let have matrix .
(a) Find the matrix of by solving for in terms of , and verify your answer by computing .
(b) Now consider with matrix . Exhibit two different vectors with the same image under . Explain why this single observation already shows has no inverse.
(c) Prove: if is a linear bijection, then its inverse map is automatically linear. (So the inverse of a linear isomorphism is again a linear map.)
Problem 3 — Coordinates as an Isomorphism
An isomorphism is a bijective linear map. If an isomorphism exists we say and are isomorphic, written — from the viewpoint of linear algebra they are “the same space in different clothes.”
Let and let be the coordinate map in the standard basis , so .
(a) Compute and . Verify is linear.
(b) Use to decide, with no polynomial algebra, whether is a basis of : map the three polynomials into and test the resulting vectors.
(c) Prove: if is an isomorphism, then is a basis of if and only if is a basis of . Conclude that isomorphic spaces have equal dimension.
Problem 4 — Counting Solutions of
This problem links the existence and uniqueness of solutions to surjectivity and injectivity of , where
(a) Row-reduce . Find and a basis for , and state the rank and nullity.
(b) ” is solvable for every ” means is surjective; ” has at most one solution for every ” means is injective. Decide which (if either) holds here, and give one with no solution and one with infinitely many.
(c) Prove: a linear map is bijective if and only if for every the equation has exactly one solution .
Problem 5 — The Derivative as a Linear Map
Differentiation is a linear map. Let be .
(a) Write the matrix of in the standard basis (apply to each basis vector and record the coordinate columns).
(b) Find and , state the nullity and rank, and verify rank-nullity (recall ). Is injective? Is it surjective onto ?
(c) Prove: for any linear map , if and only if . Use this to describe all polynomials with .
Problem 6 — Eigenvectors: Directions a Map Doesn’t Turn
A nonzero vector is an eigenvector of a linear map with eigenvalue if — the map merely stretches by the factor without rotating its direction.
(a) Find all eigenvectors and eigenvalues geometrically (no calculation with characteristic equations) for each map of : the reflection across the -axis; the projection onto the -axis; and the scaling .
(b) Let be a shear. Solve directly to find every eigenvalue and eigenvector. Explain geometrically why a shear has only a single eigendirection.
(c) Prove two facts. First, if then , so is an eigenvector of with eigenvalue . Second, for a fixed , the set (the -eigenspace) is a subspace of .
Problem 7 — Composition of Linear Maps
Linear maps can be applied one after another: the composite means “apply first, then .” Let and have matrices
(a) Compute the matrix of and the matrix of by applying each composite to the standard basis vectors. (Hint: to find a column of the matrix of , apply to a standard basis vector, then apply to the result.)
(b) Find . Using rank-nullity, explain why can never be injective — and more generally why any composite that passes through fails to be injective. Is the other composite injective?
(c) Prove: if and are linear maps and the composite is injective, then is injective. Show, using from this problem, that itself need not be injective.
Problem 8 — The Determinant of a Map: Area and Invertibility
For a matrix the determinant is . Two facts you may use: is the factor by which the map scales areas, and (as we will prove later) the map is bijective exactly when .
(a) Compute the determinant of each of
For each, state whether the map is invertible, and for any non-invertible one give its kernel.
(b) The map with matrix sends the unit square to a parallelogram. Find the area of that parallelogram. Then explain why a map with determinant must collapse the unit square onto a line segment, and connect this to the map failing to be injective.
(c) Prove directly from the definition of the kernel that a linear map with matrix is injective if and only if .
Problem 9 — An Isomorphism Between Abstract Spaces
Let be the space of real matrices and define by .
(a) Show is linear and bijective — hence an isomorphism — and use it to conclude .
(b) Let be the symmetric matrices, those with . Find a basis of , state , and identify which the space is isomorphic to.
(c) Prove that "" is transitive: if and then . (You will need that the composition of two isomorphisms is again an isomorphism — prove this as part of your argument.)
Problem 10 — Capstone: Building a Linear Map to Specification
(a) The vectors and form a basis of . Find the matrix (in the standard basis) of the unique linear map with and . Is an isomorphism?
(b) Explain why there is no linear map with and . What general principle about specifying a linear map on a dependent set does this illustrate?
(c) Prove the principle behind (a) and (b): given a basis of and any vectors (not required to be distinct or independent), there exists exactly one linear map with for all . Show further that this is an isomorphism if and only if is a basis of .