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Practice Final Exam
UCLA: Math 115A
Instructor: Jens Eberhardt
- This exam has 6 questions, for a total of 60 points.
- Please print your working and answers neatly.
- Write your solutions in the space provided showing working.
- Indicate your final answer clearly.
- You may write on the reverse of a page or on the blank pages found at the back of the booklet however these will not be graded unless very clearly indicated.
- Non programmable and non graphing calculators are allowed.
Name:
ID number:
Question Points Score
1 10 2 10 3 10
4 10 5 10 6 10
Total: 60
- Consider the vector space V = P 2 (R) with its standard ordered basis
β =
1 , x, x^2
and the linear maps
T : P 2 (R) → P 2 (R), T (f ) = f (1) + f (−1)x + f (0)x^2 S : P 2 (R) → P 2 (R), S(ax^2 + bx + c) = cx^2 + bx + a.
(a) (3 points) What is [T ]β and [S]β? Show that
[T S]β =
(b) (6 points) Compute [(T S)−^1 ]β. (c) (1 point) What is (T S)−^1 (x^2 + x + 1)?
- Consider the matrix
A =
∈ M 3 , 3 (R).
(a) (2 points) Compute the characteristic polynomial of A and determine the eigenvalues and their algebraic multiplicity. (b) (6 points) Is A is diagonalizable? If yes, compute a basis β of eigenvectors of A. (c) (2 points) Compute [LA]β , where the LA is the linear transformation given by
LA : R^3 → R^3 , v 7 → Av.
- Let S : U → V and T : V → W be linear transformations between finite dimensional vector spaces U, V and W over a field F. (a) (2 points) Let v 1 , v 2 ,... , vn ∈ V be linearly independent and λ 1 , λ 2 ,... , λn ∈ F , such that λi 6 = 0 for all 1 ≤ i ≤ n. Show that also λ 1 v 1 , λ 2 v 2 ,... , λnvn are linearly independent. (b) (4 points) Let v, w ∈ V. Show that span(v, w) = span(v, v + w). Hint: Proceed in two steps: Show that for all x ∈ V,
- x ∈ span(v, w) implies x ∈ span(v, v + w) and
- x ∈ span(v, v + w) implies x ∈ span(v, w). (c) (4 points) Assume that R(S) = N(T ), i.e. the range of S is equal to the nullspace of T. Assume furthermore that S is one-to-one and T is onto. Show that
dim V = dim U + dim W.
Hint: Use the rank-nullity formula.
- Let V be a finite dimensional vector space over a field F. Recall that
L(V, V ) = {T : V → V | T is a linear transformation}
denotes the vector space of linear transformations from V to V (also called linear operators on V ). Fix a vector v ∈ V and define Z = {T ∈ L(V, V ) | T (v) = 0}. One calls Z the annihilator of v in L(V, V ). (a) (4 points) Show that Z is a subspace of L(V, V ). (b) (2 points) Let λ ∈ F such that λ 6 = 0. Prove or disprove (by finding a counterexample) that
Z′^ = {T ∈ L(V, V ) | T (v) = λv}
is a subspace of L(V, V ). (c) (2 points) Assume that β = {v 1 ,... , vn} is an ordered basis of V , such that v 1 = v. Let T ∈ L(V, V ). Show that T ∈ Z if and only if the first column of A = [T ]β equals 0. (d) (2 points) Assuming v 6 = 0, what is dim(Z)?
This page has been left intentionally blank. You may use it as scratch paper. It will not be graded unless indicated very clearly here and next to the relevant question.
