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Montano1993 [528]
3 years ago
8

How to determine if the columns of a matrix are linearly independent?

Mathematics
1 answer:
astra-53 [7]3 years ago
8 0
Let's consider an arbitrary 2x2 matrix as an example,

\mathbf A=\begin{bmatrix}\mathbf x&\mathbf y\end{bmatrix}=\begin{bmatrix}x_1&y_1\\x_2&y_2\end{bmatrix}

The columns of \mathbf A are linearly independent if and only if the column vectors \mathbf x,\mathbf y are linearly independent.

This is the case if the only way we can make a linear combination of \mathbf x,\mathbf y reduce to the zero vector is to multiply the vectors by 0; that is,

c_1\mathbf x+c_2\mathbf y=\mathbf 0

only by letting c_1=c_2=0.

A more concrete example: suppose

\mathbf A=\begin{bmatrix}1&2\\4&8\end{bmatrix}

Here, \mathbf x=\begin{bmatrix}1\\4\end{bmatrix} and \amthbf y=\begin{bmatrix}2\\8\end{bmatrix}. Notice that we can get the zero vector by taking c_1=-2 and c_2=1:

-2\begin{bmatrix}1\\4\end{bmatrix}+\begin{bmatrix}2\\8\end{bmatrix}=\begin{bmatrix}-2+2\\-8+8\end{bmatrix}=\mathbf 0

so the columns of \mathbf A are not linearly independent, or linearly dependent.
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