I need to show that the vectors $v_1 = \langle 2, 1\rangle$ and $v_2 = \langle 4, 3\rangle$ span $\mathbb R^2$ by definition. By definition if I can write any vector in $\mathbb R^2$ as a linear combination of $v_1$ and $v_2$ then the vectors span $\mathbb R^2$. How do I show this? Here is what I have been working with:
- Let $v_x = \langle c_1, c_2\rangle$ be any vector in $\mathbb R^2$ where $c_1$ and $c_2$ are in $\mathbb R$.
- $v_x = c_1\langle 1, 0\rangle + c_2\langle 0, 1\rangle$
- Set $v_x$ = a linear combination of $v_1$ and $v_2$? How do I proceed from here?
Best Answer
What you need to prove is the following:
$$\forall\,v:=(a,b)\in\Bbb R^2\,\,\,\exists\,x,y\in\Bbb R\,\,\,s.t.\,\, v=x(2,1)+y(4,3)\Longleftrightarrow$$
$$\Longleftrightarrow \text{the linear system}\,\,\left\{\begin{array}{}2x+4y=a\\{}\\\;\;x+3y=b\end{array}\right.$$
has a solution for any $\,a,b\in\Bbb R\,$
Now, the above system always has a solution (and, in fact, a unique one for each choice of $\,a,b\in\Bbb R\,$) since the reduced coefficients matrix's determinant is $\,2\cdot 3-1\cdot 4=2\neq 0\,$ , and voilá.