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
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