Let u1, , uk be linearly independent vectors in Rn , with
Chapter 6, Problem 6.70(choose chapter or problem)
Let u1, , uk be linearly independent vectors in Rn , with k < n. Show that there are n k vectors v1, , vnk such that u1, , uk , v1, , vnk form a basis for Rn . This states that any linearly independent set of vectors in Rn is either a basis, or can be expanded into a basis by adjoining more vectors. Hint: Choose v1 in Rn but not in the span of u, ,uk . If u1, , uk , v1 span Rn , stop. Otherwise, there is some v2 in Rn but not in the span of u1, , uk , v1. If u1, ,uk , v1, v2 span Rn , stop. Otherwise continue this process.
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