Why, if we have more columns than rows, can the columns not be linearly independent?
For example, suppose I have a set of vectors $\{ v_1, v_2, v_3, v_4\}$, $\forall v_i \in \mathbb{R}^3$.
[Math] Why, if we have more columns than rows, can the columns not be linearly independent
linear algebramatricesvectors
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Best Answer
Your question is related to the dimension of a vector space. There are a few fundamental facts about this concept, such as:
If $V$ is a finite-dimensional vector space, and $U\subset V$ is a subspace, then $\dim U\leq\dim V$.
A set of $n$ linearly independent vectors spans an $n$-dimensional vector space.
Combining these two facts yields the desired proof.