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v1=(1, 1) ^t

v2=(3, -1)^t

setting up the matrix gives det≠0, thus any vector in R^n can be written as a linear combination of v1 and v2.

This is where I'm getting confused.

If the numbers in the matrix were changed so det=0, can you still right any vector in R^n as a linear combination of v1 and v2?

If det=0, this would yield a free variable. In the examples in the book, they say you can write a vector as a linear combination of other vectors even if a free variable exists.