How to tell if a set of vectors forms a basis

A basis gives us enough vectors to build every vector in a space. It also lets us do that in just one way, so the coefficients we find are unique. Let’s check whether a pair of vectors gives us both of those properties in the plane.

Check that the columns span the plane

Use $v_{1} = (1, 4)$ and $v_{2} = (0, 3)$ . To reach any point $(x, y)$ , set $a v_{1} + b v_{2} = (x, y)$ .

\begin{array}{l} a = x \\ 4 a + 3 b = y \\ b = \frac{y - 4 x}{3} \end{array}

This works for every choice of x and y, so the two vectors span the plane.

Check that the columns are independent

If the target is $(0, 0)$ , the formulas give $a = b = 0$ . There’s no other combination that makes zero, so the two vectors are independent. Together, spanning and independence tell us this pair forms a basis.

Use the basis to give a vector new coordinates

For $(2, 11)$ , the formulas give $a = 2$ and $b = 1$ . In the order $v_{1}, v_{2}$ , the coordinates of $(2, 11)$ in this basis are $(2, 1)$ .

2 (1, 4) + (0, 3) = (2, 11)

Learn mathKrista KingAugust 1, 2021math, learn online, online course, online math

How to tell if a set of vectors forms a basis

Check that the columns span the plane

Check that the columns are independent

Use the basis to give a vector new coordinates

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