How to tell if two vectors are linearly independent

Two vectors in the plane are enough to reach every point if they give us two different directions to work with. If one vector is a scalar multiple of the other, both stay on the same line. Let’s see how to tell which case we have from the components.

Set a linear combination equal to zero

For $v_{1} = (1, 4)$ and $v_{2} = (3, 2)$ , set $c_{1} v_{1} + c_{2} v_{2} = (0, 0)$ . The horizontal and vertical components give two equations.

\begin{array}{l} c_{1} + 3 c_{2} = 0 \\ 4 c_{1} + 2 c_{2} = 0 \end{array}

The first equation gives $c_{1} = - 3 c_{2}$ . Substitute that into the second equation to get $- 10 c_{2} = 0$ , so $c_{2} = 0$ and $c_{1} = 0$ . The only combination that gives zero uses two zero coefficients, so the vectors are linearly independent.

Find a nonzero combination for dependent vectors

Now use $v_{1} = (- 2, 1)$ and $v_{2} = (6, - 3)$ . The second vector is $- 3$ times the first, so $3 v_{1} + v_{2} = (0, 0)$ . The coefficients $3$ and $1$ give a nonzero choice, so the vectors are linearly dependent.

A pair that includes the zero vector is also dependent. We can give the zero vector a nonzero coefficient and still get zero.

Learn mathKrista KingJuly 31, 2021math, learn online, online course, online math

How to tell if two vectors are linearly independent

Set a linear combination equal to zero

Find a nonzero combination for dependent vectors

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