Thread: linear combinations

let

A=
1 2
1-2

b=
4
0

c=
-3
-2

a) write b as a linear combination of the column vectors a1 and a2
b) use the result from a to determine solution of linear system Ax = b. does the system have any other solutions? explain.
c) write c as a linear combination of a1 and a2.


I got part a) I think

b = 2a1 +a2 since 2 times a1 plus one a2 equals the b matrix.

so then part b)

x would be

2
1
since

2 times A is b
1

c) is where I am stuck

I cant figure out a combination for
c=
-3
-2

I feel I need to subtract a2 from a1, but then the bottom position wont be -2 if I subtract a1 by 2a2

solve the two equations:

x-2y = -2
x+2y = -3

then x will be what you multiply a1 by, and y will be what you multiply a2 by.

hint: they won't be integers.

Thanks. The TA for our class explained it the same way.

made a matrix and reduced it and got the solution.