The length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors.

By the Linear Dependence Lemma (2.21), one of the vectors in this list is in the span of the previous ones

(emphasis added)

The author is not commenting on whether one of the $u$s can be written as a linear combination of the $w$s which follow it in the list. The Linear Dependence Lemma gives that one of the vectors in this list is the span of the ones preceding it in the list, i.e., one of the following things is true:

  • $u_2$ is in the span of $\{u_1\}$
  • $u_3$ is in the span of $\{u_1,u_2\}$
  • ...
  • $u_j$ is in the span of $\{u_1,\dots,u_{j-1}\}$
  • $w_1$ is in the span of $\{u_1,\dots,u_j\}$
  • $w_2$ is in the span of $\{u_1,\dots,u_j,w_1\}$
  • ...

Since the $u$s are linearly independent, it is impossible for any $u$ to be in the span of previous vectors on the list. So it must be a $w$ which is in the span of previous vectors on the list.

Tags:

Linear Algebra

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