Question 1185219: Express the column matrix b as a linear combination of the columns of A. (Use A1, A2, and A3 respectively for the columns of A.)
A =
3 1 2
1 −1 −3
,
b = −1
3
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
This gives us two equations in the three variables p, q, and r, so we will get either no solution or an infinite family of solutions.
[1] 
[2] 
Adding the two equations eliminates q:
Solve that for r:
[3] 
At this point, since the problem asks us to express B as a linear combination of the columns of A, we can simply choose a value for p, use the last equation to find the corresponding value of r, and then use either of our two original equations to find the value of q.
Choose p=1
Then r=4(1)-2=2
And q=p-3r-3=1-6-3=-8
So one solution is
If we wanted to find parametric representation of the complete family of solutions, we could use [2] and [3] to express q in terms of p:
The family of solutions is then described by
(p,q,r) = (p,-11p+3,4p-2)
We can then verify the correctness of that parametric representation by plugging those expressions for p, q, and r in [1] and [2]:
[1]: 
[2]: 
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