So I have two problems built off very similar premises which I have no idea how to solve:
Let W be the Subspace of $\mathbb{R}^4$ consisting of vectors of the form $ x = \{x_1, x_2, x_3, x_4\}$.
Find a basis for W when the components of x satisfy the given conditions:
- $x_1 – x_2 = 0$
$x_2-2x_3 = 0$
$x_3-x_4 = 0$- $-x_1 + 2x_2 = 0 $
$x_2 + x_3 = 0$
How should I approach solving these? I have no lecturer or anything right now so any help is greatly appreciated
Best Answer
For the first case, $x_1=x_2$ and $x_3=x_4$. Moreover, $2x_3 = x_2$. This ensures that all vectors that satisfy the three equations are of the form $$\langle 2x,2x,x,x\rangle\;\forall x\in\mathbb R$$
For the second case, $x_4$ can be anything, so all vectors that satisfy those equations are of the form $$\langle2x,x,-x,y\rangle\;\forall x,y\in\mathbb R$$
This means that for the first example, the basis is $\color{red}{\langle2,2,1,1\rangle}$ while for the second, a basis (there are infinitely many valid bases) that works is $\color{red}{\langle2,1,-1,0\rangle,\langle0,0,0,1\rangle}$