Questions:
Linear Combinations
1. If you assume {v1, v2, ..., vk} ⊂ ℜn, and w∈ℜn, 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}?
2. Also, if the zero vector is included among the vectors {v1, v2, ..., vk}, why would this mean that these vectors are linearly dependent?
3. 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?