Let $W$ be a vector space over $\mathbb R$ and let $T:\mathbb R^6 \to W$ be a linear transformation such that $S = \{Te_2, Te_4, Te_6\}$ spans $W$. Wich one of the following must be true?
- (A) $S$ is a basis of $W$
- (B) $T(\mathbb R^6) \ne W$
- (C) $\{Te_1, Te_3, Te_5\}$ spans $W$
- (D) $\ker T$ contains more than one element
I'm having trouble starting this problem.
Here are my findings so far:
I tried Dimension theorem and found that $\ker(T)$ contains more than three elements so (D) is incorrect.
And taking $\dim(T(\mathbb R^6))=\dim(W)$ we get $T(\mathbb R^6)=W$ so (B) is incorrect.
I think (C) option is correct but still not able to prove it.