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

Exercise for the break

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 ${}{\mathfrak {v}}$ given by the vectors

v_{1}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\,v_{2}={\begin{pmatrix}0\\0\\1\end{pmatrix}},\,{\text{ and }}\,v_{3}={\begin{pmatrix}1\\0\\0\end{pmatrix}}

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

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 that 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 that describes the change of bases from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ .

Exercise

Determine the transformation matrix ${}M_{\mathfrak {v}}^{\mathfrak {v}}$ for the identical base change of ${}{\mathfrak {v}}$ to ${}{\mathfrak {v}}$ .

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 ${}\mathbb {C} ^{2}$ that 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

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 ${}\mathbb {R} ^{3}$ given by the vectors

v_{1}={\begin{pmatrix}2\\3\\7\end{pmatrix}},\,\,v_{2}={\begin{pmatrix}1\\-3\\4\end{pmatrix}}\,{\text{ and }}\,v_{3}={\begin{pmatrix}5\\6\\9\end{pmatrix}}

Exercise

Let ${}V$ be the vector space of the polynomials of degree ${}\leq d$ . Determine the transformation matrices between the bases ${}{\mathfrak {u}}=X^{0},X^{1},X^{2},\ldots ,X^{d}$ and ${}{\mathfrak {v}}=P_{0},P_{1},P_{2},\ldots ,P_{d}$ with ${}P_{0}=1$ and

{}P_{i}=(X-1)\cdots (X-i)\,

for ${}d=0,1,2,3,4$ .

Exercise

Let ${}V$ be the vector space of all polynomials of degree ${}\leq 3$ , with the basis ${}{\mathfrak {u}}=X^{0},X^{1},X^{2},X^{3}$ . Show that the polynomials

X^{3}+3X^{2}-X+4,\,-X^{3}+4X^{2}+5X+3,\,2X^{2}-X+1,\,3X^{3}+5X^{2}+7X-2

form a basis of ${}V$ , and determine the transformation matrices.

Exercise

Let ${}V$ be the vector space of the ${}2\times 2$ -matrices with the standard basis

{}{\mathfrak {u}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,.

Show that

{}{\mathfrak {v}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\,{\begin{pmatrix}0&1\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\,

is also a basis of ${}V$ , and determine the transformation matrices.

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

Let ${}V$ and ${}W$ be finite-dimensional ${}K$ -vector spaces. Let ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$ and ${}{\mathfrak {u}}=u_{1},\ldots ,u_{n}$ be bases of ${}V$ , and ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$ and ${}{\mathfrak {z}}=z_{1},\ldots ,z_{m}$ bases of ${}W$ . Let ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ and ${}M_{\mathfrak {z}}^{\mathfrak {w}}$ be the transformation matrices. What is the transformation matrix for the base change from the basis ${}(v_{1},0),\ldots ,(v_{n},0),(0,w_{1}),\ldots ,(0,w_{m})$ to the basis ${}(u_{1},0),\ldots ,(u_{n},0),(0,z_{1}),\ldots ,(0,z_{m})$ of the product space ${}V\times W$ ?

Exercise

Let ${}K$ be a field.

a) Show that the linear subspace ${}U\subseteq K^{3}$ generated by

{\begin{pmatrix}3\\2\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\8\\-3\end{pmatrix}}

has dimension ${}2$ .

b) Determine a basis and the dimension of the solution space ${}L\subseteq K^{3}$ of the linear equation

{}-6x_{1}+4x_{2}+5x_{3}=0\,.

c) Determine a basis and the dimension of the intersection ${}U\cap L$ .

d) Confirm Theorem 9.7 in this example.

Exercise

Show that the space of ${}n\times n$ -matrices over a field ${}K$ is the direct sum of the space of the diagonal matrices, the space of the upper triangular matrices with zero diagonal, and the space of the lower triangular matrices with zero diagonal.

Exercise

Give an example of linear subspaces ${}U_{1},U_{2},U_{3}$ in a vector space ${}V$ such that ${}V=U_{1}+U_{2}+U_{3}$ , such that ${}U_{i}\cap U_{j}=0$ for ${}i\neq j$ holds, and such that the sum is not direct.

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

Let ${}V=\operatorname {Map} \,{\left(\mathbb {R} ,\mathbb {R} \right)}$ be the vector space of all functions from ${}\mathbb {R}$ to ${}\mathbb {R}$ . Show that there exists a direct sum decomposition

{}V=G\oplus U\,,

where ${}G$ denotes the linear subspace of all even functions, and ${}U$ denotes the linear subspace of all odd functions.

Exercise

Suppose that the vector space ${}V$ is the direct sum of the linear subspaces ${}V_{1}$ and ${}V_{2}$ . Show that a linear subspace ${}U\subseteq V$ is not necessarily the direct sum of the linear subspaces ${}U\cap V_{1}$ and ${}U\cap V_{2}$ .

Exercise

Determine a direct complement of the linear subspace ${}U\subseteq \mathbb {R} ^{3}$ generated by ${}{\begin{pmatrix}-5\\4\\9\end{pmatrix}}$ and ${}{\begin{pmatrix}2\\-7\\3\end{pmatrix}}$

Exercise

Let ${}V$ be a finite-dimensional ${}K$ -vector space, and let ${}U_{1},U_{2}\subseteq V$ denote linear subspaces of the same dimension. Show that ${}U_{1}$ and ${}U_{2}$ have a common direct complement.

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 ${}\mathbb {R} ^{3}$ that 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 that 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 that describes the change of basis from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ .

Exercise (4 marks)

Let ${}K$ be a field, and let ${}V$ be a ${}K$ -vector space, with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$ . Let ${}w\in V$ be a vector with a representation