# Basis Calculator

The calculator will find a basis of the space spanned by the set of given vectors, with steps shown.

Related calculators: Linear Independence Calculator, Matrix Rank Calculator

$\mathbf{\vec{v_{1}}}$ $\mathbf{\vec{v_{2}}}$ $\mathbf{\vec{v_{3}}}$

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Find a basis of the space spanned by the set of vectors $\left\{\left[\begin{array}{c}1\\2\\3\end{array}\right], \left[\begin{array}{c}9\\12\\5\end{array}\right], \left[\begin{array}{c}5\\7\\4\end{array}\right]\right\}.$

## Solution

The basis is a set of linearly independent vectors that spans the given vector space.

There are many ways to find a basis. One of the ways is to find the row space of the matrix whose rows are the given vectors.

Thus, the basis is $\left\{\left[\begin{array}{c}1\\2\\3\end{array}\right], \left[\begin{array}{c}0\\-6\\-22\end{array}\right]\right\}$ (for steps, see row space calculator).

Another way to find a basis is to find the column space of the matrix whose columns are the given vectors.

Thus, the basis is $\left\{\left[\begin{array}{c}1\\2\\3\end{array}\right], \left[\begin{array}{c}9\\12\\5\end{array}\right]\right\}$ (for steps, see column space calculator).

If two different bases were found, they are both the correct answers: we can choose any of them, for example, the first one.

The basis is $\left\{\left[\begin{array}{c}1\\2\\3\end{array}\right], \left[\begin{array}{c}0\\-6\\-22\end{array}\right]\right\}$A.