Basis of space spanned by $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}$$$

The calculator will find a basis of the space spanned by the set of the vectors $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}$$$, with steps shown.

Related calculators: Linear Independence Calculator, Matrix Rank Calculator

Number of vectors:

Size of the vectors:

Vectors:

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

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Find a basis of the space spanned by the set of the vectors $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}.$$$

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 $$$\left\{\left[\begin{array}{c}1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}$$$ (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 $$$\left\{\left[\begin{array}{c}3\\1\\2\end{array}\right], \left[\begin{array}{c}-4\\6\\7\end{array}\right], \left[\begin{array}{c}2\\8\\9\end{array}\right]\right\}$$$ (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 $$$\left\{\left[\begin{array}{c}1\\0\\0\end{array}\right], \left[\begin{array}{c}0\\1\\0\end{array}\right], \left[\begin{array}{c}0\\0\\1\end{array}\right]\right\}.$$$A