MATH 2090  Review Problems for Exam 3  Fall 2001

1. Consider the vectors v⃗_{1} = (1, 3, 4), v⃗_{2} = (0, 1, 1), v⃗_{3} = (−1, 0, −1), v⃗_{4} = (5, 3, −2) in ℝ^{3}.

(a) Does the set of vectors {v⃗_{1}, v⃗_{2}, v⃗_{3}, v⃗_{4}} span ℝ^{3}? (b) Is this set of vectors is linearly independent?

(c) Does this set of vectors form a basis for ℝ^{3}? Explain.

2. Recall that the trace of a square matrix is the sum of the diagonal entries. Let S be the set of all 2 × 2 matrices with real entries and zero trace: S = \{[ \begin{matrix}a & b \\ c & d \end{matrix} ] : a, b, c, d real, and a + d = 0\} .

(a) Show that S is a subspace of 𝕄_{2}(ℝ), the vector space of all 2 × 2 matrices with real entries.

(b) Verify that \{[ \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] , [\begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} ]\} is a basis for S.

(c) Find the components of the matrix A = [ \begin{matrix}1 & 2 \\ 1 & −1 \end{matrix} ] with respect to the basis in part (b). In other words, express the matrix A as a linear combination of the matrices given in part (b).

3.(a) Consider the mapping T : ℝ^{3} → ℝ^{3} defined by T (x_{1}, x_{2}, x_{3}) = (x_{1} + x_{2}, x_{3}, k), where k is a scalar.

Find all values of k for which T is a linear transformation. Find the matrix of T for these values of k.

(b) If T : ℝ^{3} → ℝ is a linear transformation and T (1, 0, 0) = 1, T (1, 1, 0) = 3, T (1, 1, 1) = 5, find T (2, 4, 6).

4. Consider the linear transformation T : ℝ^{3} → ℝ^{3} defined by T (x⃗) = Ax⃗, where A = [\begin{matrix}2 & 0 & −1 \\ 2 & 3 & 1 \\ −4 & 0 & 2 \end{matrix} ].

(a) Find a basis for Ker(T), the kernel of the linear transformation T. What is the dimension of Ker(T)?

(b) Find a basis for Rng(T), the range of the linear transformation T. What is the dimension of Rng(T)?

(c) Do the rows of the matrix A form a basis for ℝ^{3}? Explain.