How can I find a basis for a subspace?

Question

Let $U$ be the subspace $\mathbb R^{5}$ defined by $U$ = {($x_1, x_2, x_3, x_4, x_5$) $\in$ $\mathbb R^{5}$: $x_1 = 3x_2$, and $x_3 = 7x_4$

How can I go about finding a basis for $U$?

Answer 1

Here's a general rule of thumb. You have added $2$ linear restrictions to a $5$ dimensional space, so you should get a $3$ dimensional subspace (in general you have to be careful the restrictions are not redundant!)

Based on the restrictions, we know that $x_1$ and $x_2$ must be related to one another, and that $x_3$ and $x_4$ must be related to one another. Perhaps you could find a basis of the form $$ \begin{pmatrix} *\\*\\0\\0\\0 \end{pmatrix}, \begin{pmatrix} 0\\0\\*\\*\\0 \end{pmatrix}, \begin{pmatrix} 0\\0\\0\\0\\* \end{pmatrix} $$

Answer 2

Informally speaking, the dimension of a basis is the minimum number of basis vectors needed to span the entire subspace. Because neither $x_2$ or $x_4$ adds a dimension to the subspace, you are now working with ${(x_1,x_3,x_5)}$, which has a maximum dimension of $3$.

You could now use the Gram-Schmidt process to get an orthogonal basis.