MATH 3510 Chapter Notes - Chapter 17: Triangular Matrix, Invertible Matrix, John Wiley & Sons
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Which expresses w as a linear combination of. Proof of theorem 4. 5. 5(a) if s is a set of vectors that spans v but is not a basis for v, then s is a linearly dependent set. Thus some vector v in s is expressible as a linear combination of the other vectors in s. by the. Plus/minus theorem (4. 5. 3b), we can remove v from s, and the resulting set s will still span v. if s is linearly independent, then s is a basis for v, and we are done. Proof of theorem 4. 5. 5(b) suppose that basis for v, then s fails to span v, so there is some vector v in v that is not in. Theorem (4. 5. 3a), we can insert v into s, and the resulting set s will still be linearly independent. If s spans v, then s is a basis for v, and we are finished.
