Linear Algebra Help. For each vector, show the weights on v1 and v2 used to generate the vector --etc?

Question

normally these are simple for me, but apparently missing 1 day of class is a big deal. here's the problem they give me

a1 = [1, 4, -2] a2 = [-2, -3, 7], b = [4, 1, h].
for what values of h is b in the lane spanned by a1 and a2

I'm not entirely clear on what a span even is. this pos book explains it with one paragraph(literally) so I'm assuming it's not as hard as I'm making it out to be. The problem also states that I shouldn't make a sketch, does that mean I shouldn't draw an array? I feel like this simple assignment just made a turn for the worst.

Answer 1

It means, for what value of h can b be represented as a multiple of a combination of a1 and a2.

ca1 + da2 = b
c<1, 4, -2> + d<-2, -3, 7> = <4, 1, h>

This give us the equations:

c - 2d = 4
4c - 3d = 1
-2c + 7d = h

Subtract four times the first equation from the second.

5d = -15
d = -3

Plug back into the first equation and solve for c.

c - 2(-3) = 4
c = 4 - 6 = -2

Now solve for h.

-2c + 7d = h
h = -2(-2) + 7(-3) = 4 - 21 = -17

Answer 2

As long as one vector can't be written as a multiple of the others, the two vectors span a plane. That means any vector
in the plane can be written as a vector sum of multiples of the
two so-called basis vectors. That is the definition of a span
So to answer the problem
write x(i+4j-2k)+y(-2i-3j+7k)=4i+j+hk
ignore the terms in the k component and try to find x and y
to make b the i and j components of b. This is impossible. b points outside the plane, not inside its span.