Is $\{V_1, V_2, V_3\}$ in $\mathbb R^4$?

Question

So given this:

I believe since there are only 3 vectors aka columns that this would be a no. I am not sure of the reasoning as there is no way for every row to have a pivot column, eliminating A.

I also can't seem to find a linear combination between these matrices. So I think that eliminates B and D, leaving C.

So would the answer be C?

I am sure I can work this out by hand to solve for RREF, but I am not sure if that is necessary to answer this question efficiently.

Answer 1

The objective is to determine whether the vectors $\{V_1,V_2,V_3\}$ spans $\mathbb{R}^4$ or not.

Note that let $A$ be a $m\times n$ matrix. Then, the column of $A$ span $\mathbb{R}^m$ if and only if $A$ has a pivot position in every row.

Let $A=[V_1\ \ V_2\ \ V_3]$ be the matrix. Then, reduce the matrix $A$ into the reduced echelon form.

If the matrix $A=[V_1\ \ V_2\ \ V_3]$ does not have a pivot position in each row, then the columns of the matrix $A$ does not span $\mathbb{R}^4$