Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 8

Exercise for the break

Exercise

Matrices/2x2/Dimension/Exercise

Exercises

Let ${}K$ be a field, and let ${}V$ be a ${}K$ -vector space of dimension ${}n=\dim _{}{\left(V\right)}$ . Suppose that ${}n$ vectors ${}v_{1},\ldots ,v_{n}$ in ${}V$ are given. Prove that the following facts are equivalent.

${}v_{1},\ldots ,v_{n}$ form a basis for ${}V$ .
${}v_{1},\ldots ,v_{n}$ form a system of generators for ${}V$ .
${}v_{1},\ldots ,v_{n}$ are linearly independent.

Exercise

Vector space/Finite dimension/Linear subspace of full dimension/Exercise

Exercise

Parametrized vectors/abc/Cyclically swapped/Dimensions/Exercise

Exercise *

Let ${}K$ be a field, and let ${}V$ and ${}W$ be two finite-dimensional ${}K$ vector spaces with

{}\dim _{}{\left(V\right)}=n\,

and

{}\dim _{}{\left(W\right)}=m\,.

What is the dimension of the Cartesian product ${}V\times W$ ?

Exercise

Linear subspaces/Sum of the dimensions larger/Intersection/Exercise

Exercise

Let ${}K$ be a field, and let ${}K[X]$ denote the polynomial ring over ${}K$ . Let ${}d\in \mathbb {N}$ . Show that the set of all polynomials of degree ${}\leq d$ is a finite dimensional subspace of ${}K[X]$ . What is its dimension?

Exercise

Show that the set of all real polynomials of degree ${}\leq 4$ , which have a zero for ${}-2$ and for ${}3$ , form a finite-dimensional linear subspace in ${}\mathbb {R} [X]$ . Determine its dimension.

Exercise

Let ${}V$ be a finite-dimensional vector space over the complex numbers, and let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$ . Prove that the family of vectors

v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}

form a basis for ${}V$ , considered as a real vector space.

Exercise

Consider the standard basis ${}e_{1},e_{2},e_{3},e_{4}$ in ${}\mathbb {R} ^{4}$ and the three vectors

{\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.

Prove that these vectors are linearly independent and extend them to a basis by adding an appropriate standard vector as shown in the base change theorem. Can one take any standard vector?

Exercise

Linear system/Basis completion/1/Exercise

Exercise

Basic multiplication table/Last digit/Vector space dimension/Exercise

Exercise

Infinite basis/Finite basis/Exercise

Magical square/Linear/Definition

In this sense, the matrix

{\begin{pmatrix}c&0&\cdots &0\\0&c&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&c\end{pmatrix}}

is, for every ${}c\in K$ , a magical square.

Exercise

Magical squares/Linear/linear subspace/Exercise

Hand-in-exercises

Exercise (2 marks)

Let ${}K$ be a field, and let ${}V$ be a ${}K$ -vector space. Let ${}v_{1},\ldots ,v_{m}$ be a family of vectors in ${}V$ , and let

{}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,

be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of ${}U$ is exactly ${}m$ .

Exercise Linear system/Dimension/2/Exercise

Linear system/Dimension/2/Exercise

Exercise (4 marks)

Show that the set of all real polynomials of degree ${}\leq 6$ , which have a zero at ${}-1$ , at ${}0$ and at ${}1$ , is a finite dimensional subspace of ${}\mathbb {R} [X]$ . Determine the dimension of this vector space.

Exercise Magical squares/Linear/Dimension/Exercise

Magical squares/Linear/Dimension/Exercise

Exercise Linear system/Basis completion/2/Exercise

Linear system/Basis completion/2/Exercise

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