Given the vectors 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.