Duality

Explain why each linear functional is surjective or is the zero map.

Give three distinct examples of linear functionals on $\mathbb {R}^{[0,1]}$.

Suppose $V$ is finite-dimensional and $v \in V$ with $v \neq 0$. Prove that there exists $\varphi \in V^{\prime }$ such that $\varphi (v) = 1$.

Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$ such that $U \neq V$. Prove that there exists $\varphi \in V^{\prime }$ such that $\varphi (u) = 0$ for every $u \in U$ but $\varphi \neq 0$.

Suppose $T \in \mathcal{L}(V, W)$ and $w_{1}, \ldots , w_{m}$ is a basis of range $T$. Hence for each $v \in V$, there exist unique numbers $\varphi_{1}(v), \ldots , \varphi_{m}(v)$ such that

thus defining functions $\varphi_{1}, \ldots , \varphi_{m}$ from $V$ to $\mathbf{F}$. Show that each of the functions $\varphi_{1}, \ldots , \varphi_{m}$ is a linear functional on $V$.

Suppose $\varphi , \beta \in V^{\prime }$. Prove that $\operatorname {null}\varphi \subseteq \operatorname {null}\beta$ if and only if there exists $c \in \mathbf{F}$ such that $\beta = c \varphi$.

Suppose that $V_{1}, \ldots , V_{m}$ are vector spaces. Prove that $\left(V_{1} \times \cdots \times V_{m}\right)^{\prime }$ and $V_{1}{ }^{\prime } \times \cdots \times V_{m}{ }^{\prime }$ are isomorphic vector spaces.

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$ and $\varphi_{1}, \ldots , \varphi_{n}$ is the dual basis of $V^{\prime }$. Define $\Gamma : V \rightarrow \mathbf{F}^{n}$ and $\Lambda : \mathbf{F}^{n} \rightarrow V$ by

Explain why $\Gamma$ and $\Lambda$ are inverses of each other.

Suppose $m$ is a positive integer. Show that the dual basis of the basis $1, x, \ldots , x^{m}$ of $\mathcal{P}_{m}(\mathbb {R})$ is $\varphi_{0}, \varphi_{1}, \ldots , \varphi_{m}$, where

Here $p^{(k)}$ denotes the $k^{\text{th }}$ derivative of $p$, with the understanding that the $0^{\text{th }}$ derivative of $p$ is $p$.

Suppose $m$ is a positive integer.

Suppose $v_{1}, \ldots , v_{n}$ is a basis of $V$ and $\varphi_{1}, \ldots , \varphi_{n}$ is the corresponding dual basis of $V^{\prime }$. Suppose $\psi \in V^{\prime }$. Prove that

Suppose $S, T \in \mathcal{L}(V, W)$.

Show that the dual map of the identity operator on $V$ is the identity operator on $V^{\prime }$.

Define $T: \mathbb {R}^{3} \rightarrow \mathbb {R}^{2}$ by

Suppose $\varphi_{1}, \varphi_{2}$ denotes the dual basis of the standard basis of $\mathbb {R}^{2}$ and $\psi_{1}, \psi_{2}, \psi_{3}$ denotes the dual basis of the standard basis of $\mathbb {R}^{3}$.

Define $T: \mathcal{P}(\mathbb {R}) \rightarrow \mathcal{P}(\mathbb {R})$ by

for each $x \in \mathbb {R}$.

Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that

Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is invertible if and only if $T^{\prime } \in \mathcal{L}\left(W^{\prime }, V^{\prime }\right)$ is invertible.

Suppose $V$ and $W$ are finite-dimensional. Prove that the map that takes $T \in \mathcal{L}(V, W)$ to $T^{\prime } \in \mathcal{L}\left(W^{\prime }, V^{\prime }\right)$ is an isomorphism of $\mathcal{L}(V, W)$ onto $\mathcal{L}\left(W^{\prime }, V^{\prime }\right)$.

Suppose $U \subseteq V$. Explain why

Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Show that

Suppose $V$ is finite-dimensional and $U$ and $W$ are subspaces of $V$.

Suppose $V$ is finite-dimensional and $U$ and $W$ are subspaces of $V$.

Suppose $V$ is finite-dimensional and $\varphi_{1}, \ldots , \varphi_{m} \in V^{\prime }$. Prove that the following three sets are equal to each other.

1. $\operatorname {span}\left(\varphi_{1}, \ldots , \varphi_{m}\right)$

2. $\left(\left(\operatorname {null}\varphi_{1}\right) \cap \cdots \cap \left(\operatorname {null}\varphi_{m}\right)\right)^{0}$

3. $\left\{ \varphi \in V^{\prime }: \left(\operatorname {null}\varphi_{1}\right) \cap \cdots \cap \left(\operatorname {null}\varphi_{m}\right) \subseteq \operatorname {null}\varphi \right\}$

Suppose $V$ is finite-dimensional and $v_{1}, \ldots , v_{m} \in V$. Define a linear map $\Gamma : V^{\prime } \rightarrow \mathbf{F}^{m}$ by $\Gamma (\varphi ) = \left(\varphi \left(v_{1}\right), \ldots , \varphi \left(v_{m}\right)\right)$.

Suppose $V$ is finite-dimensional and $\varphi_{1}, \ldots , \varphi_{m} \in V^{\prime }$. Define a linear map $\Gamma : V \rightarrow \mathbf{F}^{m}$ by $\Gamma (v) = \left(\varphi_{1}(v), \ldots , \varphi_{m}(v)\right)$.

Suppose $V$ is finite-dimensional and $\Omega$ is a subspace of $V^{\prime }$. Prove that

Suppose $T \in \mathcal{L}\left(\mathcal{P}_{5}(\mathbb {R})\right)$ and $\operatorname {null}T^{\prime } = \operatorname {span}(\varphi )$, where $\varphi$ is the linear functional on $\mathcal{P}_{5}(\mathbb {R})$ defined by $\varphi (p) = p(8)$. Prove that

Suppose $V$ is finite-dimensional and $\varphi_{1}, \ldots , \varphi_{m}$ is a linearly independent list in $V^{\prime }$. Prove that

Suppose $V$ and $W$ are finite-dimensional and $T \in \mathcal{L}(V, W)$.

Suppose $V$ is finite-dimensional and $\varphi_{1}, \ldots , \varphi_{n}$ is a basis of $V^{\prime }$. Show that there exists a basis of $V$ whose dual basis is $\varphi_{1}, \ldots , \varphi_{n}$.

Suppose $U$ is a subspace of $V$. Let $i: U \rightarrow V$ be the inclusion map defined by $i(u) = u$. Thus $i^{\prime } \in \mathcal{L}\left(V^{\prime }, U^{\prime }\right)$.

The double dual space of $V$, denoted by $V^{\prime \prime }$, is defined to be the dual space of $V^{\prime }$. In other words, $V^{\prime \prime } = \left(V^{\prime }\right)^{\prime }$. Define $\Lambda : V \rightarrow V^{\prime \prime }$ by

for each $v \in V$ and each $\varphi \in V^{\prime }$.

Suppose $U$ is a subspace of $V$. Let $\pi : V \rightarrow V / U$ be the usual quotient map. Thus $\pi^{\prime } \in \mathcal{L}\left((V / U)^{\prime }, V^{\prime }\right)$.