Answer to Question #101024 in Linear Algebra for Sean

Question

1. 1. Find a basis for the subspace of R Superscript 3 that is spanned by the vectors

v1 = (1, 0, 0), v2 = (1, 1, 0), v3 = (3, 1, 0), v4 = (0, -2, 0). Choose the correct letter.

A. v1 and v2 form a basis for span {v1, v2, v3, v4}.
B . v2 and v3 form a basis for span {v1, v2, v3, v4}.
C. v3 and v4 form a basis for span {v1, v2, v3, v4}.
D. v1 and v3 form a basis for span {v1, v2, v3, v4}.
E. v2 and v4 form a basis for span {v1, v2, v3, v4}.
F. v1 and v4 form a basis for span {v1, v2, v3, v4}.

Answer 1

$c_1v_1+c_2v_2+c_3v_3+c_4v_4=0$

$c_1(1, 0, 0)+c_2(1, 1, 0)+c_3(3, 1, 0)+c_4(0, -2, 0)=0$

From the above equation we get the following equations:

$c_1+c_2+3c_3=0$

$c_2+c_3-2c_4=0$

Simplifying we get:

$c_1=2c_3-2c_4$

$c_2=2c_4-c_3$

The {$v_1,v_2$ } spans span of subspace and are linearly independent.

Indeed,

$c_1v_1+c_2v_2=0,$ $c_1(1,0,0)+c_2(1,1,0)=0.$

We obtain the following equations

$c_1+c_2=0,$

$c_2=0.$

Hence $c_2=0, c_1=0.$

Thus, {$v_1,v_2$ } is the basis.

Checking option (A)

$v_3=2v_1+v_2,$ $v_4=2v_1-2v_2.$

Thus, $v_1$ and $v_2$ form a basis for span $\{ v_1, v_2, v_3, v_4 \}.$