Question

Determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1=[-1 3 -1],a2=[-2 -3 6],b=[-6 9 2]

Answer 1

Answer:

Yes

Step-by-step explanation:

A vector is a quantity that has magnitude and direction.

We say that a vector u spans vectors v, w if u = av + bw where a and b are scalars.

Given:

$a_1=\left ( -1, 3 ,-1 \right ),a_2=\left ( -2, -3, 6 \right ),b=\left ( -6, 9, 2 \right )$

We need to write vector b as a linear combination of vectors $a_1,a_2$.

Let c and d be scalars.

$b=ca_1+da_2\\\left ( -6, 9, 2 \right )=c\left ( -1, 3 ,-1 \right )+d\left ( -2, -3, 6 \right )\\\left ( -6, 9, 2 \right )=\left ( -c-2d,3c-3d,-c+6d \right )\\-6=-c-2d\,\,...(i)\,,\,9=3c-3d\,\,...(ii)\,,\,2=-c+6d\,\,...(iii)b=ca_1+da_2$

On solving equations (i) and (ii), we get $c=4\,,\,d=1$

Now, we will check if $c=4\,,\,d=1$ satisfy equation (iii).

$-c+6d=-4+6(1)=-4+6=2$

As it satisfy this equation, so b is a linear combination of $a_1,a_2$