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MATH UCLA Linear Algebra Questions

MATH UCLA Linear Algebra Questions

1. Prove or disprove the following statements. (a) [3 pts] Let V be the subspace of P2(R) given by V = {a + bx + cx2
| a + b + c = 0}. There exists an
isomorphism T : V → R
3 (b) [3 pts] Every linear map S : M2×2(C) → C
4
is an isomorphism.2. Let T : P1(R) → P1(R) be the linear map defined by T(f(x)) = f(2)x + 3f
0
(x). Let p(x) = 4 + 6x and
γ = {1, x}. (a) [3 pts] Compute T(p(x)) using the definition of T. Then find [T(p(x))]γ. (b) [3 pts] Find [T]
γ
γ. (c) [4 pts] Compute [p(x)]γ and then use matrix multiplication to compute [T]
γ
γ[p(x)]γ. Verify that it equals
[T(p(x))]γ.
3. Let T be the linear operator defined in Problem 2. (a) [3 pts] Find all eigenvalues of T. You can use what you computed in Problem 2. (b) [5 pts] Find a basis β of eigenvectors for T. What is [T]
β
β
? (c) [3 pts] Is there a basis α for P1(R) such that [T]
α
α =
12 4
0 5
? Justify your answer. 4. Suppose V be a vector space over a field F with dim(V ) = n. Let α, β, and γ be ordered bases for V . Let P
denote the matrix that changes α coordinates into β coordinates, and let Q denote the matrix that changes
β coordinates to γ coordinates. (a) [4 pts] Prove that Q−1
is the matrix that changes γ coordinates into β coordinates. (b) [4 pts] Recall the linear functions φα : V → F
n and φγ : V → F
n are given by φα(x) = [x]α and
φγ(x) = [x]γ. Prove that LQLP φα = φγ.

Quadratic Functions and Transformations Standard and Vertex Form Problems

Quadratic Functions and Transformations Standard and Vertex Form Problems

The test will include the following lessons:❑ Chapter 4.1 – Quadratic functions and transformations
❑ Chapter 4.2 – Standard and Vertex form of a quadratic function
❑ Chapter 4.4 – Factoring Quadratic Expressions
❑ Chapter 4.7 – The Quadratic Formula and Discriminant
❑ Chapter 4.8 – Complex Numbers
❑ Chapter 4.9 – Quadratic Systems

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