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MATH UCLA Linear Algebra Questions

MATH UCLA Linear Algebra Questions

Question Description

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 anisomorphism T : V ? R3

(b) [3 pts] Every linear map S : M2×2(C) ? C4is an isomorphism.

2. Let T : P1(R) ? P1(R) be the linear map defined by T(f(x)) = f(2)x + 3f0(x). Let p(x) = 4 + 6x and? = {1, x}.

(a) [3 pts] Compute T(p(x)) using the definition of T. Then find [T(p(x))]?.

(b) [3 pts] Find [T]??.

(c) [4 pts] Compute [p(x)]? and then use matrix multiplication to compute [T]??[p(x)]?. Verify that it equals[T(p(x))]?.

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 ? of eigenvectors for T. What is [T]???

(c) [3 pts] Is there a basis ? for P1(R) such that [T]?? = 12 40 5 ? Justify your answer.

4. Suppose V be a vector space over a field F with dim(V ) = n. Let ?, ?, and ? be ordered bases for V . Let Pdenote the matrix that changes ? coordinates into ? coordinates, and let Q denote the matrix that changes? coordinates to ? coordinates.

(a) [4 pts] Prove that Q?1is the matrix that changes ? coordinates into ? coordinates.

(b) [4 pts] Recall the linear functions ?? : V ? Fn and ?? : V ? Fn are given by ??(x) = [x]? and??(x) = [x]?. Prove that LQLP ?? = ??.

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