Base change/Transformation matrix/Sums/Exercise

Let ${}{\mathfrak {v}}=v_{1},v_{2},v_{3}$ be a basis of a three-dimensional ${}K$-vector space ${}V$.

a) Show that ${}{\mathfrak {w}}=v_{1},v_{1}+v_{2},v_{2}+v_{3}$ is also a basis of ${}V$.

b) Determine the transformation matrix ${}M_{\mathfrak {v}}^{\mathfrak {w}}$.

c) Determine the transformation matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}$.

d) Compute the coordinates with respect to the basis ${}{\mathfrak {v}}$ for the vector, which has the coordinates ${}{\begin{pmatrix}4\\8\\-9\end{pmatrix}}$ with respect to the basis ${}{\mathfrak {w}}$.

e) Compute the coordinates with respect to the basis ${}{\mathfrak {w}}$ for the vector, which has the coordinates ${}{\begin{pmatrix}3\\-7\\5\end{pmatrix}}$ with respect to the basis ${}{\mathfrak {v}}$.

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