Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 9

Exercise for the break

Exercise

Base change/Transformation matrix/R^3/Standard basis and permuted standard basis/Exercise

Exercises

Exercise

We consider the families of vectors

{\mathfrak {v}}={\begin{pmatrix}7\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\1\end{pmatrix}}\,\,{\text{  and }}\,\,{\mathfrak {u}}={\begin{pmatrix}4\\6\end{pmatrix}},\,{\begin{pmatrix}7\\3\end{pmatrix}}

in ${}\mathbb {R} ^{2}$ .

a) Show that ${}{\mathfrak {v}}$ and ${}{\mathfrak {u}}$ are both a basis of ${}\mathbb {R} ^{2}$ .

b) Let ${}P\in \mathbb {R} ^{2}$ denote the point which has the coordinates ${}(-2,5)$ with respect to the basis ${}{\mathfrak {v}}$ . What are the coordinates of this point with respect to the basis ${}{\mathfrak {u}}$ ?

c) Determine the transformation matrix which describes the change of bases from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ .

Exercise

Base change/Same Basis/Identity matrix/Exercise

Exercise

Determine the transformation matrices ${}M_{\mathfrak {v}}^{\mathfrak {u}}$ and ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ , for the standard basis ${}{\mathfrak {u}}$ , and the basis ${}{\mathfrak {v}}$ of ${}\mathbb {C} ^{2}$ , which is given by the vectors

v_{1}={\begin{pmatrix}3+5{\mathrm {i} }\\1-{\mathrm {i} }\end{pmatrix}}{\text{ and }}v_{2}={\begin{pmatrix}2+3{\mathrm {i} }\\4+{\mathrm {i} }\end{pmatrix}}.

Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$ -vector space of dimension ${}n$ . Let ${}{\mathfrak {u}}=u_{1},\ldots ,u_{n},\,{\mathfrak {v}}=v_{1},\ldots ,v_{n}$ and ${}{\mathfrak {w}}=w_{1},\ldots ,w_{n}$ denote bases of ${}V$ . Show that the transformation matrices fulfill the relation

{}M_{\mathfrak {w}}^{\mathfrak {u}}=M_{\mathfrak {w}}^{\mathfrak {v}}\circ M_{\mathfrak {v}}^{\mathfrak {u}}\,.

Exercise

Direct product/Base change/1/Exercise

Exercise

K^3/Example for intersection and sum/Bases/1/Exercise

Exercise

Matrix space/Direct sum of column space/Exercise

Exercise

Matrix space/Diagonal matrices/Direct sum/Exercise

Exercise

Pairwisely direct sum/No direct sum/Exercise

A function ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ is called even, if for all ${}x\in \mathbb {R}$ , the identity

{}f(x)=f(-x)\,

holds.

A function ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ is called odd, if for all ${}x\in \mathbb {R}$ , the identity

{}f(x)=-f(-x)\,

holds.

Exercise

R to R/Even and odd/Direct sum/Exercise

Exercise

Vector space/Direct Sum/Linear subspace not/Exercise

Exercise

K^3/Plane/Direct complement/1/Exercise

Exercise

Linear subspaces/Same dimension/Common direct complement/Exercise

Hand-in-exercises

Exercise (4 marks)

Determine the transformation matrices ${}M_{\mathfrak {v}}^{\mathfrak {u}}$ and ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ , for the standard basis ${}{\mathfrak {u}}$ , and the basis ${}{\mathfrak {v}}$ of ${}\mathbb {R} ^{3}$ , which is given by the vectors

v_{1}={\begin{pmatrix}4\\5\\1\end{pmatrix}},\,\,v_{2}={\begin{pmatrix}2\\3\\-8\end{pmatrix}}\,{\text{ and }}\,v_{3}={\begin{pmatrix}5\\7\\-3\end{pmatrix}}

Exercise (6 (3+1+2) marks)

We consider the families of vectors

{\mathfrak {v}}={\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}4\\7\\1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\5\end{pmatrix}}\,\,{\text{  and }}\,\,{\mathfrak {u}}={\begin{pmatrix}0\\2\\4\end{pmatrix}},\,{\begin{pmatrix}6\\6\\1\end{pmatrix}},\,{\begin{pmatrix}3\\5\\-2\end{pmatrix}}

in ${}\mathbb {R} ^{3}$ .

a) Show that ${}{\mathfrak {v}}$ and ${}{\mathfrak {u}}$ are both a basis of ${}\mathbb {R} ^{3}$ .

b) Let ${}P\in \mathbb {R} ^{3}$ denote the point which has the coordinates ${}(2,5,4)$ with respect to the basis ${}{\mathfrak {v}}$ . What are the coordinates of this point with respect to the basis ${}{\mathfrak {u}}$ ?

c) Determine the transformation matrix which describes the change of basis from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ .

Exercise Base change/Exchange lemma/Exercise

Base change/Exchange lemma/Exercise

Exercise K^4/Example for intersection and sum/Bases/2/Exercise

K^4/Example for intersection and sum/Bases/2/Exercise

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