21-341  Linear Algebra I  Fall 2003

Review Problems for Test 1

1. Let 𝔽 = ℝ and V = ℝ^{4}. Determine whether or not the list of vectors is linearly independent.

(a) < 1, 1, 2, 1 >, < 1, 1, 2, 2 >

(b) < 1, −1, 1, 0 >, < 1, 2, 3, 4 >, < 1, 2, 3, 1 >, < −1, 1, −1, 1 >, < 1, 0, 1, 0 >

(c) < 1, −1, 1, 1 >, < 0, 0, 0, 0 >, < 1, 3, 2, 1 >

(d) < 1, 2, −1, 0 >, < 2, 1, 0, 1 >, < 4, 5, −2, 4 >

2. Let 𝔽 = ℝ and V = ℝ^{5}. Find a basis for the solution set of the equation x_{1} + 2x_{2} − x_{3} + x_{4} − 2x_{5} = 0.

3. Prove or Disprove: Let V be a finitely generated vector space and let S, T, R be subspaces of V. Then S + (T ∩ R) = (S + T) ∩ (S + R).

4. Let 𝔽 = ℝ and let n ∈ ℤ^{+} be given. Let P_{n}(ℝ) denote the vector space of all real polynomials of the degree ≤ n. Assume that f_{0}, f_{1}, … f_{n} ∈ P_{n}(ℝ) satisfy f_{0}(π) = f_{1}(π) = f_{2}(π) = … = f_{n}(π) = 0. Show that the list f_{0}, f_{1}, f_{2}, …, f_{n} is linearly dependent.

5. Let 𝔽 be a field and V, W be vector spaces over 𝔽. Let L : V → W be a mapping such that L(u + v) = L(u) + L(v) and L(λu) = λL(u) for all u, v ∈ V, λ ∈ 𝔽. Let

T = {L(u) : u ∈ V }.

Show that T is a subspace of W.

6. Let 𝔽 = ℝ and let P_{3}(ℝ) be the vector space of all real polynomials of degree

≤ 3. Let S = {f ∈ ℙ_{3}(ℝ) : f(1) = 2∫_{0}^{1}f(x)dx}.

Show that S is a subspace of P_{3}(ℝ) and find a basis for S.

7. Let 𝔽 = ℝ and A = ( \begin{matrix}1 & 2 & 1 & −1 \\ 1 & 5 & 2 & 1 \\ −2 & 5 & 5 & 4 \end{matrix} ).

Find a 3 × 2 matrix A′ such that A′ is row equivalent to A, all zero rows of A′ are below the nonzero rows, and the nonzero rows of A′ are in echelon form.

8. Let V be an eight dimensional vector space and let S, T be subspaces of V with dim(S) = 5 and dim(T) = 6. What is the smallest possible dimension of S ∩ T.

9. Prove or Disprove: Let S, T, R be subspaces of a finitely generated vector space V. If S + T = S + R then T = R.

10. In this problem, we will write complex numbers in the form a+bi where a, b ∈ ℝ and i^{2} = −1; for such a number we write

Re(a + bi) = a, Im(a + bi) = b.

Let V = ℂ^{4}.

(a) Assume that 𝔽 = ℝ. Which of the following are subspaces?

i. {< z_{1}, z_{2}, z_{3}, z_{4} >: z_{1} + z_{4} = 0} ii. {< z_{1}, z_{2}, z_{3}, z_{4} >: z_{1} + z_{4} = 1} iii. {< z_{1}, z_{2}, z_{3}, z_{4} >: z_{1} + z_{4} = z_{2}} iv. {< z_{1}, z_{2}, z_{3}, z_{4} : Re(z_{1}) = 0} v. {< z_{1}, x_{2}, z_{3}, z_{4} >: Im(z_{1}) = 0} vi. {< z_{1}, z_{2}, z_{3}, z_{4} >: z_{1}z_{2} = 0}

(b) How would the answers to part (a) change if the field were changed from ℝ to ℂ.

11. Let 𝔽 = ℝ and V = F(ℝ) the set of all real-valued functions f : ℝ → ℝ. Let f_{1}(x) = e^{x}, f_{2}(x) = e^{2x}, f_{3}(x) = e^{3x}. Determine whether or not f_{1}, f_{2}, f_{3} are linearly independent.