Finding a Basis for a Subspace Defined by an Equation

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In Chapter 1 we called them "free variables". Here I'll describe a subspace containing free variables, and show you how to represent it algebraically, as well as how to express a basis for such a space.
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OOPS! At about 2:40, I said, and wrote, 4 instead of 8!

leslieglen
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Can we take y and z as free variables instead of x and y?

sulemanali
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How do we conclude the 2 basis vectors are linearly independent by reducing to echelon form in the last step?

liteandsound