Basis Calculator

Find bases of a vector space step by step

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

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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find a basis of the space spanned by the set of the 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 $$$\left\{\left[\begin{array}{c}1\\0\\- \frac{13}{3}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{11}{3}\end{array}\right]\right\}$$$ (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.

Answer

The basis is $$$\left\{\left[\begin{array}{c}1\\0\\- \frac{13}{3}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{11}{3}\end{array}\right]\right\}\approx \left\{\left[\begin{array}{c}1\\0\\-4.333333333333333\end{array}\right], \left[\begin{array}{c}0\\1\\3.666666666666667\end{array}\right]\right\}.$$$A