Given a vector space V, a spanning subset and a linearly independent set, why is the spanning set larger then the LI set?

Question

This is taken from my Linear Algebra 1 course, and the proof involves forming a matrix. How do I prove that the spanning set is equal to or larger then the linearly independent set?

Answer 1

Depending on what you have already seen (and proven) about these concepts, there is not much to be done. If the given vector space $V$ is $k$-dimensional, then:

• any linearly independent set contains at most $k$ vectors;
• any spanning set (for $V$) contains at least $k$ vectors.

From this, what you want immediately follows.

The fact that your text uses a matrix to show this can either mean you don't have the results from above (yet), are not allowed to use them or it could simply be a choice/preference for this matrix approach.

This is taken from my Linear Algebra 1 course, and the proof involves forming a matrix. How do I prove that the spanning set is equal to or larger then the linearly independent set?

But your text gives a proof (?) so what is your question exactly; you don't understand the proof? Then it's useful to provide the proof and indicate what you don't understand.