Question

Determine whether the set of vectors 2%2C-4%29%20and%20v_%7B3%20%3D%281%2C1%2C-1%29" id="TexFormula1" title=" v_{1=(3,2,1), v_{2} =(-1,-2,-4) and v_{3 =(1,1,-1)" alt=" v_{1=(3,2,1), v_{2} =(-1,-2,-4) and v_{3 =(1,1,-1)" align="absmiddle" class="latex-formula"> will span R3?

Answer 1

Since each vector is a member of $\mathbb R^3$, the vectors will span $\mathbb R^3$ if they form a basis for $\mathbb R^3$, which requires that they be linearly independent of one another.

To show this, you have to establish that the only linear combination of the three vectors $c_1\mathbf v_1+c_2\mathbf v_2+c_3\mathbf v_3$ that gives the zero vector $\mathbf0$ occurs for scalars $c_1=c_2=c_3=0$.

$c_1\begin{bmatrix}3\\2\\1\end{bmatrix}+c_2\begin{bmatrix}-1\\-2\\-4\end{bmatrix}+c_3\begin{bmatrix}1\\1\\-1\end{bmatrix}=(0,0,0)\iff\begin{bmatrix}3&-1&1\\2&-2&1\\1&-4&-1\end{bmatrix}\begin{bmatrix}c_1\\c_2\\c_3\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$

Solving this, you'll find that $c_1=c_2=c_3=0$, so the vectors are indeed linearly independent, thus forming a basis for $\mathbb R^3$ and therefore they must span $\mathbb R^3$.