PROF. A. SUCIU  MTH 1230 – LINEAR ALGEBRA  FALL 1997

SAMPLE QUIZ 3

1. Let A = [\begin{matrix}1 & 2 & 3 & 4 \\ 0 & 1 & 1 & 3 \\ 0 & 1 & 1 & 4 \end{matrix}] . Find:

(a) A basis for the row space of A.

(b) A basis for the column space of A.

(c) A basis for the null space of A.

(d) The rank of A.

2. In each of parts (a) and (b), are the given vectors v⃗_{1}, v⃗_{2}, v⃗_{3} independent or dependent? If they are independent, say why. If they are dependent, exhibit a linear dependence relation among them.

(a)  v⃗_{1} = [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}]  v⃗_{2} = [\begin{matrix}2 \\ 1 \\ −3 \end{matrix}]  v⃗_{3} = [\begin{matrix}−1 \\ 2 \\ −6 \end{matrix}]

(b)  v⃗_{1} = [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}]  v⃗_{2} = [\begin{matrix}2 \\ 1 \\ −3 \end{matrix}]  v⃗_{3} = [\begin{matrix}−1 \\ 2 \\ 6 \end{matrix}]

3. Which of the given subsets of the vector space, M_{23}, of all 2 × 3 matrices are subspaces? In each case, give a reason for your answer.

(a) All matrices of the form [\begin{matrix}a & b & c \\ d & 0 & 0 \end{matrix}], where b = a + c.

(b) All matrices of the form [\begin{matrix}a & b & c \\ d & 0 & 0 \end{matrix}], where c > 0.

(c) All matrices of the form [\begin{matrix}a & b & c \\ d & e & f \end{matrix}], where a = −2c and f = e + 2d.

(d) All matrices of the form [\begin{matrix}a & b & c \\ d & e & f \end{matrix}], where a = 2c + 1.

4. Which of the following sets of vectors span ℝ^{4}? In each case, give a reason for your answer.

(a) v⃗_{1} = [1 2 5 7], v⃗_{2} = [1 1 −4 0], v⃗_{3} = [0 0 0 3].

(b) v⃗_{1} = [1 0 0 1], v⃗_{2} = [0 1 0 0], v⃗_{3} = [1 1 1 1], v⃗_{4} = [1 1 1 0].

(c) v⃗_{1} = [1 1 0 0], v⃗_{2} = [0 0 1 1], v⃗_{3} = [1 1 1 1], v⃗_{4} = [2 2 −1 −1].

(d) v⃗_{1} = [3 2 1 0], v⃗_{2} = [1 0 0 0], v⃗_{3} = [0 1 7 1], v⃗_{4} = [0 0 1 2], v⃗_{5} = [0 6 1 0].