1. ## 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)

2. ## 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+y-6z = -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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3. ## 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?

4. ## 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".