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Linear Algebra: Companion notes

Abhay Soman

Exercises 2.3 Exercises

1.

Show that the set \({\rm Seq(F)}\) of all sequences with values in a field \(F\) is a vector space under the componentwise addition
\begin{equation*} (a_n)+(b_n)=(a_n+b_n) \end{equation*}
and the scalar multiplication
\begin{equation*} \alpha\cdot(x_n)=(\alpha x_n)\quad\text{where } \alpha,x_n\in F\text{ for every } n. \end{equation*}

2.

Show that the set \({\rm Seq_0(\R)}\) of all real-valued sequences converging to \(0\) is a vector space. Use the same operations defined in an earlier exercise.

3.

Show that the set \(\Q(\sqrt{2})=\{a+b\sqrt{2}:a,b\in\Q\}\) is a vector space over \(\Q\text{.}\) The addition is given by
\begin{equation*} (a+b\sqrt{2})+(c+d\sqrt{2})=(a+c)+(b+d)\sqrt{2} \end{equation*}
and the scalar multiplication by
\begin{equation*} \alpha\cdot(a+b\sqrt{2})=\alpha a+\alpha b\sqrt{2} \quad\text{for }\alpha,a,b\in\Q. \end{equation*}

4.

Let \(F\) be a field. Show that the set \(L=\{(a,a,\ldots,a)\in F^n:a\in F\}\) is a vector space over \(F\text{.}\)

5.

Show that the set \(V=\{(a,a+b)\in\R^2:a,b\in\R\}\) is a vector space over \(\R\text{.}\)

6.

Is the set \(C[0,1]\) of all real-valued continuous functions on \([0,1]\) a vector space over \(\Q\text{?}\) Justify your answer.

7.

Let \(P_1,P_2\) be two planes in \(\R^3\) passing through the origin \((0,0,0)\text{.}\) Is \(P_1\cap P_2\) a vector space over \(\R\text{?}\) Justify your answer.

8.

Is the set \(\{(a,b):a,b\in\R\text{ and }a,b\leq 0\}\) a vector space over \(\R\text{?}\) Justify your answer.

9.

The transpose of a matrix \(A\) is denoted by \(A^t\text{.}\) Is the set \({\rm Skew}_n(F)=\{A\in M_{n\times n}(F):A=-A^t\}\) a vector space over \(F\text{?}\) Justify your answer.

10.

For \(n\geq 1\text{,}\) is \(\GL_n(F)\) a vector space over \(F\text{?}\) Justify your answer.
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