# Independent Subset of Vectors

Korey Hinton -> Blog -> Math -> Linear Algebra -> Independent Subset of Vectors

## For a given set of vectors, determine whether or not they are linearly dependent. If they are dependent, find the largest possible independent subset of vectors.

Find the largest possible number of independent vectors among:

$v_{1} = \begin{bmatrix} 1\\ -1\\ 0\\ 0\\ \end{bmatrix} \; \; v_{2} = \begin{bmatrix} 1\\ 0\\ -1\\ 0\\ \end{bmatrix} \; \; v_{3} = \begin{bmatrix} 1\\ 0\\ 0\\ -1\\ \end{bmatrix} \; \; v_{4} = \begin{bmatrix} 0\\ 1\\ -1\\ 0\\ \end{bmatrix} \; \; v_{5} = \begin{bmatrix} 0\\ 1\\ 0\\ -1\\ \end{bmatrix} \; \; v_{6} = \begin{bmatrix} 0\\ 0\\ 1\\ -1\\ \end{bmatrix}$

### Step 1: combine vectors as columns of matrix A

$A = \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0\\ -1 & 0 & 0 & 1 & 1 & 0\\ 0 & -1 & 0 & -1 & 0 & 1\\ 0 & 0 & -1 & 0 & -1 & -1 \end{bmatrix}$

### Step 2: reduce to reduce row echelon form

• R1 + R2
• R2 + R3
• R3 + R4
• -R3 + R2
• -R3 + R1
• -R2 + R1
$R = \begin{bmatrix} 1 & 0 & 0 & -1 & -1 & 0\\ 0 & 1 & 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

The pivot columns (Column 1, 2, and 3) are also linearly independent and they are the ones we keep. Since not all columns are linearly independent this is a dependent set. We can throw out columns 4-6. So the original v1, v2, and v3 are the largest possible independent subset of vectors.

Date: 2014-06-28T04:59+0000

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