Basis of space spanned by $$$\left\{\left[\begin{array}{c}1\\-1\\0\\1\end{array}\right], \left[\begin{array}{c}4\\1\\4\\0\end{array}\right]\right\}$$$

Find a basis of the space spanned by the set of the vectors $$$\left\{\left[\begin{array}{c}1\\-1\\0\\1\end{array}\right], \left[\begin{array}{c}4\\1\\4\\0\end{array}\right]\right\}$$$.

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 $$$\left\{\left[\begin{array}{c}1\\0\\\frac{4}{5}\\\frac{1}{5}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{4}{5}\\- \frac{4}{5}\end{array}\right]\right\}$$$ (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 $$$\left\{\left[\begin{array}{c}1\\-1\\0\\1\end{array}\right], \left[\begin{array}{c}4\\1\\4\\0\end{array}\right]\right\}$$$ (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 $$$\left\{\left[\begin{array}{c}1\\0\\\frac{4}{5}\\\frac{1}{5}\end{array}\right], \left[\begin{array}{c}0\\1\\\frac{4}{5}\\- \frac{4}{5}\end{array}\right]\right\} = \left\{\left[\begin{array}{c}1\\0\\0.8\\0.2\end{array}\right], \left[\begin{array}{c}0\\1\\0.8\\-0.8\end{array}\right]\right\}.$$$A