Prof. A. Suciu  MTH 1230 — LINEAR ALGEBRA  Fall 1997

QUIZ 3

1. 13 points Let A = [\begin{matrix}2 & 1 & 1 & 0 \\ 4 & 2 & 2 & 0 \\ −6 & −3 & 0 & 1 \\ 2 & 1 & −2 & 0 \end{matrix}].

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

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

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

2. 7 points Let A be a 12 × 20 matrix. Suppose that A = L · U, where L is a lower triangular matrix, with 1’s down the diagonal, and U is a row-echelon matrix, with 9 non-zero pivots.

(a) Find rank A.

(b) Find dim RS(A).

(c) Find dim CS(A).

(d) Find dim NS(A).

(e) Is RS(A) = RS(U)?  YES  NO  MAYBE
(f) Is CS(A) = CS(U)?  YES  NO  MAYBE
(g) Is NS(A) = NS(U)?  YES  NO  MAYBE

MTH 1230  Quiz 3  Fall 1997

3. 8 points Are the following vectors independent or dependent? If they are independent, say why. If they are dependent, exhibit a linear dependence relation among them.

v⃗_{1} = [\begin{matrix}2 \\ 2 \\ 6 \end{matrix}]  v⃗_{2} = [\begin{matrix}3 \\ −1 \\ 5 \end{matrix}]  v⃗_{3} = [\begin{matrix}−5 \\ 7 \\ −3 \end{matrix}]

4. 12 points In each of the following, a vector space V and a subset S are given. Circle one answer:

(a) V = ℝ^{3}, S = {(t, 2t, 3t) | t is a real number}

S is closed under addition:  YES  NO  MAYBE
S is closed under scalar multiplication:  YES  NO  MAYBE
S is a vector subspace of V:  YES  NO  MAYBE

(b) V = ℝ^{3}, S = {(t, 2t, 3t) | t is a positive real number}

S is closed under addition:  YES  NO  MAYBE
S is closed under scalar multiplication:  YES  NO  MAYBE
S is a vector subspace of V:  YES  NO  MAYBE

(c) V = ℝ^{3}, S = {(t, 2t, 3t) | t is an integer}

S is closed under addition:  YES  NO  MAYBE
S is closed under scalar multiplication:  YES  NO  MAYBE
S is a vector subspace of V:  YES  NO  MAYBE

(d) V = ℝ^{3}, S = {(t + 1, 2t, 3t − 1) | t is a real number}

S is closed under addition:  YES  NO  MAYBE
S is closed under scalar multiplication:  YES  NO  MAYBE
S is a vector subspace of V:  YES  NO  MAYBE