Finding the basis of a subspace in $\mathbb R^4$

Question

Find a basis of the subspace of ${\mathbb R}^4$ consisting of all vectors of the form $$ \left\lbrack \begin{array}{c} x_1 \\ 2 x_1 + x_2 \\ 6 x_1 + 2 x_2 \\ 8 x_1 - 4 x_2 \end{array} \right\rbrack $$ The answer should be a list of row vectors.

Answer 1

You should post your attempts, so we know where in the process you're struggling.

Here's a worked out solution, please comment so we can help you learn instead of just giving you an answer.

Notice that $\begin{bmatrix}x_1\\2x_1+x_2\\6x_1+2x_2\\8x_1-4x_2 \end{bmatrix}$.

Recall that a basis is a set of vectors such that a linear combination of them can form every vector in the subspace.

We can rewrite the subspace as $x_1\begin{bmatrix}1\\2\\6\\8 \end{bmatrix}+x_2\begin{bmatrix}0\\1\\2\\-4 \end{bmatrix}$.

What does this reveal? This directly reveals the basis vectors, because by definition, any $x_1v_1+x_2v_2$ IS a linear combination of $v_1$ and $v_2$.