Can a matrix be a vector?

They are not vectors in the form that you usually encounter them (i.e. as a string of coordinates in n-dimensional space).
However, if you call any element of a vector space (that is, a set that satisfies certain axioms) a vector, then indeed they are.

Note by the way that there is always a bijection between n x m matrices and vectors in R^nm.

 

You don't need the determinant to solve this problem. You are just looking for some a, b such that [itex]a\,\mathbf v_1 + b\,\mathbf v_2 = \mathbf v_3[/itex] -- or show that no such a, b exist.

 

Nope. Edit: Oops, my bad. I checked it again after seeing DH's post, and I'm the one who messed up.

I suggest you use the standard definition of linear independence. [itex]\{v_1,\dots,v_k\}[/itex] is linearly independent if

[tex]\sum a_i v_i=0\Rightarrow a_i=0 \text{ for all }i[/tex]

(What DH said works too).

 

Good. So what does that mean in terms of the problem? (You need to determine whether v1, v2, v3 are dependent or independent.)

That looks good.

 

Correct.