Math 54 quiz 4 let v a1 a2 ai isin r be the vector

Question

MATH 54 QUIZ 4-

1. Let P₄ be the vector space of polynomials of degree at most 4, and let V be the subset of polynomials of degree exactly 4. Is V a subspace of P4? Explain your reasoning.

2. Let M_2×2 be the vector space of 2 × 2 matrices. Let H be the subset of M_2×2 consisting of matrices A such that det A = 0. Is H a subspace of M_2×2? Explain your reasoning.

3. Find the coordinates of relative to the basis B = of R³.

4. Let P₂ be the vector space of polynomials of degree at most 2. It is given that B = {1 + t, t + t², 1 + t²} is a basis for P₂. Find the coordinates of t² relative to B.

5. Suppose that T : V → W is an injective (one-to-one) linear transformation. Let {v₁, . . . , v_n} be a linearly independent set in V . Prove that {T(v₁), . . . , T(v_n)} linearly independent.

6. Let V = {(a₁, a₂, . . .) : a_i ∈ R} be the vector space of infinite sequences of real numbers. The right shift operator is a linear transformation T : V → V defined by T((a₁, a₂, . . .)) = (0, a₁, a₂, . . .). Find the kernel and range of T.

7. Write anything you like here (comments, questions, suggestions, etc).