Extension to a basis
Task number: 2539
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 \(V=\mathbb R^{2\times 2}\).