1. Label the following statements as true or false. Assume that all vector spaces are finite-dimensional.

   1. (a) Every linear transformation is a linear functional.

   2. (b) A linear functional defined on a field may be represented as a $1\times 1$ matrix.

   3. (c) Every vector space is isomorphic to its dual space.

   4. (d) Every vector space is isomorphic to the dual of some vector space.

   5. (e) If T is an isomorphism from V onto V* and $\beta$ is a finite ordered basis for V, then $\text{T}(\beta )={\beta }^{*}$.

   6. (f) If T is a linear transformation from V to W, then the domain of ${({\text{T}}^t)}^t$ is V**.

   7. (g) If V is isomorphic to W, then V* is isomorphic to W*.

   8. (h) The derivative of a function may be considered as a linear functional on the vector space of differentiable functions.

2. For the following functions f on a vector space V, determine which are linear functionals.

   1. (a) $\text{V}=\text{P}(R);\text{f}(p(x))=2p^{\prime }(0)+p^{″}(1)$ where′ denotes differentiation

   2. (b) $\text{V}={\text{R}}^2;\text{f}(x,y)=(2x,4y)$

   3. (c) $\text{V}={\text{M}}_{2\times 2}(F);\text{f}(A)=\text{tr}(A)$

   4. (d) $\text{V}={\text{R}}^3;\text{f}(x,y,z)=x^2+y^2+z^2$

   5. (e) $\text{V}=\text{P}(R);\text{f}(p(x))={\displaystyle {\int }_0^{1}p(t)dt}$

   6. (f) $\text{V}={\text{M}}_{2\times 2}(F);\text{f}(A)=A_{11}$

3. For each of the following vector spaces V and bases $\beta$, find explicit formulas for vectors of the dual basis ${\beta }^{*}$ for V*, as in Example 4.

   1. (a) $\text{V}={\text{R}}^3;\beta =\{(1,0,1),(1,2,1),(0,0,1)\}$

   2. (b) $\text{V}={\text{P}}_2(R);\beta =\{1,x,x^2\}$

4. Let $\text{V}={\text{R}}^3$, and define ${\text{f}}_1,{\text{f}}_2,{\text{f}}_3\in {\text{V}}^{*}$ as follows:

   $${\text{f}}_1(x,y,z)=x-2y,{\text{ f}}_2(x,y,z)=x+y+z,{\text{ f}}_3(x,y,z)=y-3z.$$

   Prove that $\{{\text{f}}_1,{\text{f}}_2,{\text{f}}_3\}$ is a basis for V*, and then find a basis for V for which it is the dual basis.

5. Let $\text{V}={\text{P}}_1(R)$, and, for $p(x)\in \text{V}$, define ${\text{f}}_1,{\text{f}}_2\in {\text{V}}^{*}$ by

   $${\text{f}}_1(p(x))=\underset{0}{\overset{1}{{\displaystyle \int }}}p(t)dt{\text{ and f}}_2(p(x))=\underset{0}{\overset{2}{{\displaystyle \int }}}p(t)dt.$$

   Prove that $\{{\text{f}}_1,{\text{f}}_2\}$ is a basis for V*, and find a basis for V for which it is the dual basis.

6. Define $\text{f}\in {({\text{R}}^2)}^{*}$ by $\text{f}(x,y)=2x+y$ and $\text{T}:{\text{R}}^2\to {\text{R}}^2$ by $\text{T}(x,y)=(3x+2y,x)$.

   1. (a) Compute ${\text{T}}^t(\text{f})$.

   2. (b) Compute ${{\displaystyle [{\text{T}}^t]}}_{{\beta }^{*}}$, where $\beta$ is the standard ordered basis for ${\text{R}}^2$ and ${\beta }^{*}=\{{\text{f}}_1,{\text{f}}_2\}$ is the dual basis, by finding scalars a, b, c, and d such that ${\text{T}}^t({\text{f}}_1)=a{\text{f}}_1+c{\text{f}}_2$ and ${\text{T}}^t({\text{f}}_2)=b{\text{f}}_1+d{\text{f}}_2$.

   3. (c) Compute ${{\displaystyle [\text{T}]}}_{\beta }$ and ${\left({{\displaystyle [\text{T}]}}_{\beta }\right)}^t$, and compare your results with (b).

7. Let $\text{V}={\text{P}}_1(R)$ and $\text{W}={\text{R}}^2$ with respective standard ordered bases $\beta$ and $\gamma$. Define $\text{T}:\text{V}\to \text{W}$ by

   $$\text{T}(p(x))=(p(0)-2p(1),p(0)+p^{\prime }(0)),$$

   where $p^{\prime }(x)$ is the derivative of p(x).

   1. (a) For $\text{f}\in {\text{W}}^{*}$ defined by $\text{f}(a,b)=a-2b$, compute ${\text{T}}^t(\text{f})$.

   2. (b) Compute ${{\displaystyle [{\text{T}}^t]}}_{{\gamma }^{*}}^{{\beta }^{*}}$ without appealing to Theorem 2.25.

   3. (c) Compute ${{\displaystyle [\text{T}]}}_{\beta }^{\gamma }$ and its transpose, and compare your results with (b).

8. Let $\{u,v\}$ be a linearly independent set in ${\text{R}}^3$. Show that the plane $\{su+tv:s,t\in R\}$ through the origin in ${\text{R}}^3$ may be identified with the null space of a vector in ${\left({\text{R}}^3\right)}^{*}$.

9. Prove that a function $\text{T}:{\text{F}}^n\to {\text{F}}^m$ is linear if and only if there exist ${\text{f}}_1,{\text{f}}_2,\dots ,{\text{f}}_m\in {({\text{F}}^n)}^{*}$ such that $\text{T}(x)=({\text{f}}_1(x),{\text{f}}_2(x),\dots ,{\text{f}}_m(x))$ for all $x\in {\text{F}}^n$. Hint: If T is linear, define ${\text{f}}_i(x)=(g_i\text{T})(x)$ for $x\in {\text{F}}^n$; that is, ${\text{f}}_i={\text{T}}^t(g_i)$ for $1\le i\le m$, where $\{g_1,g_2,\dots ,g_m\}$ is the dual basis of the standard ordered basis for ${\text{F}}^m$.

10. Let $\text{V}={\text{P}}_n(F)$, and let $c_0,c_1,\dots ,c_n$ be distinct scalars in F.

    1. (a) For $0\le i\le n$, define ${\text{f}}_i\in {\text{V}}^{*}$ by ${\text{f}}_i(p(x))=p(c_i)$. Prove that $\{{\text{f}}_0,{\text{f}}_1,\dots ,{\text{f}}_n\}$ is a basis for V*. Hint: Apply any linear combination of this set that equals the zero transformation to $p(x)=(x-c_1)(x-c_2)\cdots (x-c_n)$, and deduce that the first coefficient is zero.

    2. (b) Use the corollary to Theorem 2.26 and (a) to show that there exist unique polynomials $p_0(x),p_1(x),\dots ,p_n(x)$ such that $p_i(c_j)={\delta }_{ij}$ for $0\le i\le n$. These polynomials are the Lagrange polynomials defined in Section 1.6.

    3. (c) For any scalars $a_0,a_1,\dots ,a_n$ (not necessarily distinct), deduce that there exists a unique polynomial q(x) of degree at most n such that $q(c_i)=a_i$ for $0\le i\le n$. In fact,

       $$q(x)={\displaystyle \sum }_{i=0}^n{a}_i{p}_i(x).$$

    4. (d) Deduce the Lagrange interpolation formula:

       $$p(x)={\displaystyle \sum _{i=0}^{n}p(c_i)p_i(x)}$$

       for any $p(x)\in \text{V}$.

    5. (e) Prove that

       $$\underset{a}{\overset{b}{{\displaystyle \int }}}p(t)dt={\displaystyle \sum }_{i=0}^{n}p(c_i)d_i,$$

       where

       $$d_i=\underset{a}{\overset{b}{{\displaystyle \int }}}{p}_i(t)dt.$$

       Suppose now that

       $$c_i=a+{\displaystyle \frac{i(b-a)}{n}}\text{ for }i=0,1,\dots ,n.$$

       For $n=1$, the preceding result yields the trapezoidal rule for evaluating the definite integral of a polynomial. For $n=2$, this result yields Simpson’s rule for evaluating the definite integral of a polynomial.

11. Let V and W be finite-dimensional vector spaces over F, and let ${\psi }_1$ and ${\psi }_2$ be the isomorphisms between V and V** and W and W**, respectively, as defined in Theorem 2.26. Let $\text{T}:\text{V}\to \text{W}$ be linear, and define ${\text{T}}^{tt}={({\text{T}}^t)}^t$. Prove that the diagram depicted in Figure 2.6 commutes (i.e., prove that ${\psi }_2\text{T}={\text{T}}^{tt}{\psi }_1$). Visit goo.gl/Lkd6XZ for a solution.

    

    Figure 2.6 Full Alternative Text

12. Let V be a finite-dimensional vector space with the ordered basis $\beta$. Prove that $\psi (\beta )={\beta }^{**}$, where $\psi$ is defined in Theorem 2.26.

In Exercises 13 through 17, V denotes a finite-dimensional vector space over F. For every subset S of V, define the annihilator $S^0$ of S as

13. 1. (a) Prove that $S^0$ is a subspace of V*.

    2. (b) If W is a subspace of V and $x\notin \text{W}$, prove that there exists $\text{f}\in {\text{W}}^0$ such that $\text{f}(x)\ne 0$.

    3. (c) Prove that ${(S^0)}^0=\text{span}(\psi (S))$, where $\psi$ is defined as in Theorem 2.26.

    4. (d) For subspaces ${\text{W}}_1$ and ${\text{W}}_2$ , prove that ${\text{W}}_1={\text{W}}_2$ if and only if ${{\displaystyle \text{W}}}_1^0={{\displaystyle \text{W}}}_2^0$.

    5. (e) For subspaces ${\text{W}}_1$ and ${\text{W}}_2$, show that ${({\text{W}}_1+{\text{W}}_2)}^0={{\displaystyle \text{W}}}_1^0\cap {{\displaystyle \text{W}}}_2^0$.

14. Prove that if W is a subspace of V, then $\text{dim}(\text{W})+\text{dim}({\text{W}}^0)=\text{dim}(\text{V})$. Hint: Extend an ordered basis $\{x_1,x_2,\dots ,x_k\}$ of W to an ordered basis $\beta =\{x_1,x_2,\dots ,x_n\}$ of V. Let ${\beta }^{*}=\{{\text{f}}_1,{\text{f}}_2,\dots ,{\text{f}}_n\}$. Prove that $\{{\text{f}}_{k+1},{\text{f}}_{k+2},\dots ,{\text{f}}_n\}$ is a basis for ${\text{W}}^0$.

15. Suppose that W is a finite-dimensional vector space and that $\text{T}:\text{V}\to \text{W}$ is linear. Prove that $\text{N}({\text{T}}^t)={(\text{R}(\text{T}))}^0$.

16. Use Exercises 14 and 15 to deduce that $\text{rank}({\text{L}}_{A^t})=\text{rank}({\text{L}}_A)$ for any $A\in {\text{M}}_{m\times n}(F)$.

In Exercises 17 through 20, assume that V and W are finite-dimensional vector spaces. (It can be shown, however, that these exercises are true for all vector spaces V and W.)

17. Let T be a linear operator on V, and let W be a subspace of V. Prove that W is T-invariant (as defined in the exercises of Section 2.1) if and only if ${\text{W}}^0$ is ${\text{T}}^t$-invariant.

18. Let V be a nonzero vector space over a field F, and let S be a basis for V. (By the corollary to Theorem 1.13 (p. 61) in Section 1.7, every vector space has a basis.) Let $\text{Φ}:{\text{V}}^{*}\to \mathcal{F}(S,F)$ be the mapping defined by $\text{Φ}(\text{f})={\text{f}}_S$, the restriction of f to S. Prove that $\text{Φ}$ is an isomorphism. Hint: Apply Exercise 35 of Section 2.1.

19. Let V be a nonzero vector space, and let W be a proper subspace of V (i.e., $\text{W}\ne \text{V}$).

    1. (a) Let $\text{g}\in {\text{W}}^{*}$ and $v\in \text{V}$ with $v\notin \text{W}$. Prove that for any scalar a there exists a function $\text{f}\in {\text{V}}^{*}$ such that $\text{f}(v)=a$ and $\text{f}(x)=\text{g}(x)$ for all x in W. Hint: For the infinite-dimensional case, use Exercise 4 of Section 1.7 and Exercise 35 of Section 2.1.

    2. (b) Use (a) to prove there exists a nonzero linear functional $\text{f}\in {\text{V}}^{*}$ such that $\text{f}(x)=0$ for all $x\in \text{W}$.

20. Let V and W be nonzero vector spaces over the same field, and let $\text{T}:\text{V}\to \text{W}$ be a linear transformation.

    1. (a) Prove that T is onto if and only if ${\text{T}}^t$ is one-to-one.

    2. (b) Prove that ${\text{T}}^t$ is onto if and only if T is one-to-one.

    Hint: In the infinite-dimensional case, use Exercise 19 for parts of the proof.