
Linear algebra help
Can someone explain why b can't be written as a linear combination of v1, v2, and v3? (These are all column vectors.)
I get infinite solutions when I try to solve the equation
c1v1 + c2v2 + c3v3 = b
but they don't actually work when I plug them in.
v1 = (1, 2, 0)
v2 = (0, 1, 2)
v3 = (5, 6, 8)
b = (2, 1, 6)

Re: Linear algebra help
If you calculate the determinant of [v1 v2 v3] it equals zero, which means the values are not linearly independent. It actually reduces to the equations:
2x+y6z = 1
Y+4z= 3
Y+4z=3
Since the last two equations are the same this means there a infinite number of solutions. For example both (2,3,0) and (3,1, 1) are solutions.
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Re: Linear algebra help
That's what I found...an infinite number of solutions.
So if my textbook says b is not a linear combination of the other vectors, can I just assume it's a typo?

Re: Linear algebra help
Is that exacty what your textbooks says? b clearly can be written as a linear combination of the given vector, but not uniquely: b cannot be written as a "unique linear combination of v1, v2, and v3".