## Extension to a basis

Extend the set $$M$$ to a basis of the vector space $$V$$

• #### Variant 1

$$M=\{(1{,}2,0{,}0)^T,(2{,}1,1{,}3)^T,(0{,}1,0{,}1)^T\}$$, $$V=\mathbb R^4$$.

• #### Variant 2

$$M=\{-x^2,x+x^2,x^3-1\}$$, in the space $$V$$ of real polynomes of degree at most three.

• #### Variant 3

$$M=\left\{ \begin{pmatrix} 0 & 3\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 0\\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1\\ 2 & 0 \end{pmatrix} \right\}$$ in the space $$V=\mathbb R^{2\times 2}$$.