Solve the Equation.

Question

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 \\ \end{Bmatrix}$$

How can i solve this ?

I found it $$v_1+v_2=0$$ $$v_1+v_2=0$$ .

So i can't solve it for $v_1$ and $v_2$ .

Answer 1

You essentially have $1$ equation in two unknowns. Let $v_2= t$, where $t\in \mathbb R$. Then $v_1 = -v_2 = -t$.

So there are infinitely many solutions, all of which can be represented by $$V = \begin{pmatrix} -t\\t\end{pmatrix} = t\begin{pmatrix} -1 \\ 1\end{pmatrix}$$ again, where $t$ can take any value in the reals (assuming that's the field on which $V$ is defined.)

NOTE: You could just as easily set $t = v_1 \implies v_2 = -t = -v_1$.

Answer 2

There are infinite solutions. For every $v_1$ there is a $v_2$ which is negative of it.

Answer 3

because you have only two variables $v_1, v_2$ you can think of the equation $v_1 + v_2 = 0$ as a line through the origin with slope $-1.$ you have one line in the plane because the second equation does not anything more to this line. any point on this line is a solution. all solutions are given by $v_1 = t, v_2 = -t$ where you can take $t$ to be anything.