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So c has to be a "scalar."

To find these c values you can set up the augmented matrix (v1, v2, v3, v) and find the RREF. I'm a little confused. If you have a free variable in in the RREF matrix, that means c1, c2, and c3 aren't "scalars." does that mean v is not a linear combination of v1, v2, v3?

It would seem so from that question.

However in the next section in my textbook, it shows that you can write v as a linear combination of v1, v2, v3 even if you have a free variable in the augmented matrix. Isn't that contradictory?