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(1) Solve the system of linear equations:
x + y − z = 5 z + w = − 2 2 x + 2y − z + w = 8 x + y + w = 3
Describe the solution set as a set of vectors. Is it a subspace of R^4? Justify your answer.
(2) Now consider the homogeneous system associated to the system in Problem 1:
x + y − z = 0 z + w = 0 2 x + 2y − z + w = 0 x + y + w = 0
(a) Describe the solution set, W , as the span of a set S 1 and conclude that W is a subspace of R^4. (b) Does the set
S 2 =
span W? Is it linearly independent? (c) Find a subset S 3 of S 2 which is a basis of W. (d) Finally, complete S 3 to a basis B of R^4.
(3) Solve the system of equations { ax + y = a^2 x + ay = 1
where a ∈ R is a parameter, and x and y are the unknowns. For which values of a does the system have no solutions? What about infinitely many solutions?
(4) Determine whether the following sets S are subspaces of R^4. If S is a subspace, find a basis and determine the dimension of that subspace.
1
(a) S = {
x y z w
|^ x^ −^3 y^ +^ z^ = 0,^2 x^ + 5w^ = 0}.
(b) S = {
x y z w
|^ x^ +^ y^ +^ z^ = 0, x^ +^ y^ +^ w^ = 0,^ −x^ −^ y^ −^3 z^ + 2w^ = 0}.
(c) S = {
x y z w
|^ xy^ = 0}.
(5) Consider the following subset of the vector space M 2 × 2 of 2 by 2 matrices:
W =
{[
a b c d
] ∣∣
∣ 2 a^ −^ c^ −^ d^ = 0 and^ a^ + 3b^ = 0
(a) Is W a subspace of M 2 × 2? (b) Find a set S of matrices such that W = span(S). (c) Is the set S you just found basis for W? If not, then find a basis for W. (d) Complete the basis of W from part (c) to a basis B of M 2 × 2.
(6) Let h : P 2 → R be a homomorphism from the space of polynomials of degree at most 2 to R. Suppose that h(x − 1) = 1, h(x^2 − 1) = 3 and h(5x) = 10. Calculate h(x^2 + x − 7).
(7) Determine the dimension of the space of homomorphisms between P 5 , the space of polynomials with real cofficients of degree at most 5, and M 2 , 3 , the space of 2 × 3 matrices with real entries. Are the two vector spaces isomorphic? If so, find an explicit isomorphism.
(8) True or false? If you answer “true” you must give a proof; if you answer “false” you must give a counterexample. (a) Between any n-dimensional vector space V and Rn^ there exists exactly one iso- morphism. (b) A matrix can represent two different linear maps. (c) The set of integers Z is a subspace of R. (d) A system with more equations than unknowns cannot have infinitely many solu- tions. (e) A system with more unknowns than equations has at least one solution. (f) If a set of vectors v 1 ,... , vr is linearly independent and c is a non-zero scalar, then the set of vectors cv 1 ,... , cvr is also linearly independent. (g) Let V and W be two vector spaces. Then the space of isomorphisms from V to W is a vector space.
