《线性代数应该这样学》课内验证2-有限维向量空间

CHAPTER 2 Finite-Dimensional Vector Spaces

1. If some vectors are removed from a linearly independent list, the remaining
   list is also linearly independent, as you should verify.

Proof Suppose is linearly independent, then the only choice of that makes is .

Since the order of vectors does not affect linear independence, we put all the vectors that need to be removed on the right.

Now assume the vectors are removed, and the remaining is linearly dependent.

Hence there exist , not all , such that .

Let all are , hence , at the same time , not all . We get a contradiction.

Thus the remaining lise is also linearly independent.