Question 1 Which set of vectors is a basis of $\mathbb{R}^2$ ? $\{ \left( \begin{smallmatrix} -2 \\ -15 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 \\ 0 \end{smallmatrix} \right)\}$ $\{ \left( \begin{smallmatrix} 25 \\ 144 \end{smallmatrix} \right), \left( \begin{smallmatrix} 5 \\ 12 \end{smallmatrix} \right)\}$ $\{ \left( \begin{smallmatrix} -11 \\ -7 \end{smallmatrix} \right), \left( \begin{smallmatrix} -132 \\ -84 \end{smallmatrix} \right)\}$ $\{ \left( \begin{smallmatrix} -5 \\ -15 \end{smallmatrix} \right), \left( \begin{smallmatrix} 3 \\ -5 \end{smallmatrix} \right)\}$
Question 2 What are the components of $\vec{c}= \left( \begin{smallmatrix} 1 \\ -5 \\ 4 \\ \end{smallmatrix} \right)$ in the basis $K$ ? $(0,3,9)$ $(3, -4, 1)$ $(-3, 4, 2)$ $(-5,1,7)$
Question 3 How is called a basis having all its vectors of length $1$ and being perpendicular to each other ? normal basis canonical basis standard basis orthonormal basis
Question 4 Can any basis include the null vector ? no yes, if the determinant of its vectors is different to $0$ yes, if its are linearly dependent yes, if the determinant of its vectors equals $0$
Question 5 Let $\vec{m}= \left( \begin{smallmatrix} -3 \\ 4 \\ 2 \\ \end{smallmatrix} \right)$ expressed with respect to the basis $G$. What are its components in the standard basis ? $(2,-1,-3)$ $(-5,-3,1)$ $(0,-1,-1)$ $(1,-5,4)$
Question 6 Why must the vectors of a basis be linearly independent ? for a change basis into the standard basis to be possible because the determinant of them must be equal to $0$ because any vector of the related space must be generated in a unique way because the determinant of them must be different to $0$