Abstract Algebra: Theory and Applications

If $S=\{v_1,\dots ,v_n\}$ is a set of linearly independent vectors for $V\text{,}$ then $S$ is a basis for $V\text{.}$

If $S=\{v_1,\dots ,v_n\}$ spans $V\text{,}$ then $S$ is a basis for $V\text{.}$

Prove that the kernel of $T\text{,}$ $\ker (T)=\{v\in V:T(v)=\mathbf{0}\}\text{,}$ is a subspace of $V\text{.}$ The kernel of $T$ is sometimes called the null space of $T\text{.}$

Prove that the range or range space of $T\text{,}$ $R(V)=\{w\in W:T(v)=w\text{ for some }v\in V\}\text{,}$ is a subspace of $W\text{.}$

Show that $T:V\to W$ is injective if and only if $\ker (T)=\{\mathbf{0}\}\text{.}$

Let $\{v_1,\dots ,v_k\}$ be a basis for the null space of $T\text{.}$ We can extend this basis to be a basis $\{v_1,\dots ,v_k,v_{k+1},\dots ,v_m\}$ of $V\text{.}$ Why? Prove that $\{T(v_{k+1}),\dots ,T(v_m)\}$ is a basis for the range of $T\text{.}$ Conclude that the range of $T$ has dimension $m-k\text{.}$

Let $\dim V=\dim W\text{.}$ Show that a linear transformation $T:V\to W$ is injective if and only if it is surjective.

Prove that $U+V$ and $U\cap V$ are subspaces of $W\text{.}$

If $U+V=W$ and $U\cap V=\mathbf{0}\text{,}$ then $W$ is said to be the direct sum. In this case, we write $W=U\oplus V\text{.}$ Show that every element $w\in W$ can be written uniquely as $w=u+v\text{,}$ where $u\in U$ and $v\in V\text{.}$

Let $U$ be a subspace of dimension $k$ of a vector space $W$ of dimension $n\text{.}$ Prove that there exists a subspace $V$ of dimension $n-k$ such that $W=U\oplus V\text{.}$ Is the subspace $V$ unique?

Let $V$ be an $F$-vector space. Define the dual space of $V$ to be $V^{\ast }=\mathrm{Hom}(V,F)\text{.}$ Elements in the dual space of $V$ are called linear functionals. Let $v_1,\dots ,v_n$ be an ordered basis for $V\text{.}$ If $v={\alpha }_1{v}_1+\cdots +{\alpha }_n{v}_n$ is any vector in $V\text{,}$ define a linear functional ${\varphi }_i:V\to F$ by ${\varphi }_i(v)={\alpha }_i\text{.}$ Show that the ${\varphi }_i$’s form a basis for $V^{\ast }\text{.}$ This basis is called the dual basis of $v_1,\dots ,v_n$ (or simply the dual basis if the context makes the meaning clear).

Consider the basis $\{(3,1),(2,-2)\}$ for ${\mathbb{R}}^2\text{.}$ What is the dual basis for $({\mathbb{R}}^2{)}^{\ast }\text{?}$

Let $V$ be a vector space of dimension $n$ over a field $F$ and let $V^{\ast \ast }$ be the dual space of $V^{\ast }\text{.}$ Show that each element $v\in V$ gives rise to an element ${\lambda }_v$ in $V^{\ast \ast }$ and that the map $v\mapsto {\lambda }_v$ is an isomorphism of $V$ with $V^{\ast \ast }\text{.}$