Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 7

Exercise for the break

Exercise

Give an example of three vectors in ${}\mathbb {R} ^{3}$ such that each two of them is linearly independent, but all three vectors together are linearly dependent.

Exercises

Exercise

Q^2/Non-trivial representation of 0/1/Exercise

Exercise

Q^3/Non-trivial representation of 0/1/Exercise

Exercise

Show that the three vectors

{\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}

in ${}\mathbb {R} ^{4}$ are linearly independent.

Exercise

Linearly independent/Basis in linear subspace/Exercise

Exercise

Determine a basis for the solution space of the linear equation

{}3x+4y-2z+5w=0\,.

Exercise

Determine a basis for the solution space of the linear system of equations

-2x+3y-z+4w=0{\text{ and }}3z-2w=0.

Exercise

Let ${}K$ be a field. Find a linear system of equations in three variables, whose solution space is exactly

{\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.

Exercise

Linear subspaces/Intersection/(2 1 7), (4 -2 9) and (3 1 0), (5 2 -4)/Exercise

Exercise

Let ${}K$ be a field, let ${}V$ be a ${}K$ -vector space and let ${}v_{i}$ , ${}i\in I$ , be a family of vectors in ${}V$ . Prove the following facts.

If the family is linearly independent, then for each subset ${}J\subseteq I$ , also the family ${}v_{i}$ , ${}i\in J$ is linearly independent.
The empty family is linearly independent.
If the family contains the null vector, then it is not linearly independent.
If a vector appears several times in the family, then the family is not linearly independent.
A vector ${}v$ is linearly independent if and only if ${}v\neq 0$ .
Two vectors ${}v$ and ${}u$ are linearly independent if and only if ${}u$ is not a scalar multiple of ${}v$ and vice versa.

Exercise

Linear subspace/Q^n/Integer basis/Exercise

Let ${}K$ be a field, let ${}V$ be a ${}K$ -vector space, and let ${}v_{i}$ , ${}i\in I$ be a family of vectors in ${}V$ . Let ${}\lambda _{i}$ , ${}i\in I$ be a family of elements ${}\neq 0$ in ${}K$ . Prove that the family ${}v_{i}$ , ${}i\in I$ , is linearly independent (a system of generators of ${}V$ , a basis of ${}V$ ), if and only if the same holds for the family ${}\lambda _{i}v_{i}$ , ${}i\in I$ .

Hand-in-exercises

Exercise (2 marks)

Establish if in ${}\mathbb {R} ^{3}$ the three vectors

{\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}

form a basis.

Exercise (2 marks)

Establish if in ${}\mathbb {C} ^{2}$ the two vectors

{\begin{pmatrix}2-7{\mathrm {i} }\\-3+2{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6{\mathrm {i} }\\3-17{\mathrm {i} }\end{pmatrix}}

form a basis.

Exercise Matrices/Standard matrices/Basis/Exercise

Matrices/Standard matrices/Basis/Exercise

Exercise (4 marks)

Let ${}\mathbb {Q} ^{n}$ be the ${}n$ -dimensional standard vector space over ${}\mathbb {Q}$ , and let ${}v_{1},\ldots ,v_{n}\in \mathbb {Q} ^{n}$ be a family of vectors. Prove that this family is a ${}\mathbb {Q}$ -basis of ${}\mathbb {Q} ^{n}$ if and only if the same family, considered as a family in ${}\mathbb {R} ^{n}$ , is a ${}\mathbb {R}$ -basis of ${}\mathbb {R} ^{n}$ .

Exercise (3 marks)

Let ${}K$ be a field, and let

{}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}\,

be a nonzero vector. Find a linear system of equations in ${}n$ variables with ${}n-1$ equations, whose solution space is exactly

{\left\{\lambda {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\mid \lambda \in K\right\}}.

Exercise Polynomial ring/Basis/Linear form product/Exercise

Polynomial ring/Basis/Linear form product/Exercise

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