How to test linear independence in three dimensions

In three dimensions, we need three independent directions to reach every point. Even if the first two vectors form a plane, the third has to take us out of that plane. We can check that by looking for combinations of the vectors that give zero.

Use the vectors as columns

Let $v_{1} = (1, 2, 3)$ , $v_{2} = (0, 1, 4)$ , and $v_{3} = (2, - 3, 0)$ . To solve $c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3} = 0$ , put their components into the columns of an augmented matrix.

[\begin{array}{r} 1 & 0 & 2 & | & 0 \\ 2 & 1 & - 3 & | & 0 \\ 3 & 4 & 0 & | & 0 \end{array}]

Look for a pivot in every column

Subtract $2 R_{1}$ from $R_{2}$ and $3 R_{1}$ from $R_{3}$ . Then subtract $4 R_{2}$ from $R_{3}$ using the new second row.

[\begin{array}{r} 1 & 0 & 2 & | & 0 \\ 0 & 1 & - 7 & | & 0 \\ 0 & 0 & 22 & | & 0 \end{array}]

The third row gives $c_{3} = 0$ . The other two rows then give $c_{2} = c_{1} = 0$ . The vectors are linearly independent and span three-dimensional space.

Use the same test when there are more vectors

With four vectors in three dimensions, the matrix has four variable columns and only three rows. At least one variable is free, so the vectors are dependent.

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

How to test linear independence in three dimensions

Use the vectors as columns

Look for a pivot in every column

Use the same test when there are more vectors

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