If you assume {v1, v2, ..., vk} and , and you also assume {v1, v2, ..., vk} are linearly independent and {v1, v2, ..., vk, w} are linearly dependent. How would you show that w can be uniquely expressed as a linear combination of {v1, v2, ..., vk}?
Also, if the zero vector is included among the vectors {v1, v2, ..., vk}, why would this mean that these vectors are linearly dependent?
Also, if w is a linear combination of {v1, v2, ..., vk}and each vi is a linear combination of {u1, u2, ..., up}; would this make w a linear combination of {u1, u2, ..., up}, and why?
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