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Chapter 2: Problem 3

Which of the following pairs of vector spaces are isomorphic? Justify your answers. (a) \(\mathrm{F}^{3}\) and \(\mathrm{P}_{3}(F)\). (b) \(\mathrm{F}^{4}\) and \(\mathrm{P}_{3}(F)\). (c) \(\mathrm{M}_{2 \times 2}(R)\) and \(\mathrm{P}_{3}(R)\). (d) $\mathrm{V}=\left\\{A \in \mathrm{M}_{2 \times 2}(R): \operatorname{tr}(A)=0\right\\}\( and \)\mathrm{R}^{4}$.

Short Answer

Expert verified
(a) \(F^3\) and \(P_3(F)\) are not isomorphic. (b) \(F^4\) and \(P_3(F)\) are isomorphic. (c) \(M_{2\times 2}(R)\) and \(P_3(R)\) are isomorphic. (d) \(V\) and \(R^4\) are not isomorphic.

Step by step solution

01

(a) Comparing dimensions of \(F^3\) and \(P_3(F)\)

For the first pair, we have the vector space \(F^3\) and the vector space of polynomials \(P_3(F)\). The dimension of \(F^3\) is 3, as we have three slots in which we can place elements in the field \(F\). The vector space \(P_3(F)\) consists of polynomials of degree at most 3 over the field \(F\). The basis of \(P_3(F)\) is given by \(\{1, x, x^2, x^3\}\), so its dimension is 4. Since the dimensions of these two vector spaces are different, they are not isomorphic.
02

(b) Comparing dimensions of \(F^4\) and \(P_3(F)\)

For the second pair, we have the vector space \(F^4\) and again the vector space of polynomials \(P_3(F)\). The dimension of \(F^4\) is 4, as we have four slots in which we can place elements in the field \(F\). As mentioned before, the dimension of \(P_3(F)\) is also 4. Since the dimensions of these two vector spaces are the same, they are isomorphic.
03

(c) Comparing dimensions of \(M_{2\times 2}(R)\) and \(P_3(R)\)

For the third pair, we have the vector space of \(2\times 2\) matrices \(M_{2\times 2}(R)\) and the vector space of polynomials \(P_3(R)\). There are four entries in a \(2\times 2\) matrix, so the dimension of \(M_{2\times 2}(R)\) is 4. The dimension of \(P_3(R)\) is also 4, as discussed previously. Since the dimensions of these two vector spaces are the same, they are isomorphic.
04

(d) Comparing dimensions of \(V\) and \(R^4\)

For the last pair, we have the vector space \(V\) which consists of \(2\times 2\) matrices with trace 0 and the vector space \(R^4\). The dimension of \(R^4\) is 4. For \(V\), since the trace is 0, we have \(a_{11}=-a_{22}\), which leaves us with three free entries: \(a_{11}\), \(a_{12}\), and \(a_{21}\). Therefore, the dimension of \(V\) is 3. Since the dimensions of these two vector spaces are different, they are not isomorphic. To summarize: (a) \(F^3\) and \(P_3(F)\) are not isomorphic. (b) \(F^4\) and \(P_3(F)\) are isomorphic. (c) \(M_{2\times 2}(R)\) and \(P_3(R)\) are isomorphic. (d) \(V\) and \(R^4\) are not isomorphic.

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Most popular questions from this chapter

Find real numbers \(x, y, z\) such that \(A\) is Hermitian, where \(A=\left[\begin{array}{ccc}3 & x+2 i & y i \\ 3-2 i & 0 & 1+z i \\ y i & 1-x i & -1\end{array}\right]\)

Let \(V\) be the solution space of an \(n\) th-order homogeneous linear differential equation with constant coefficients. Fix \(t_{0} \in R\), and define a mapping \(\Phi: V \rightarrow \mathrm{C}^{n}\) by $$ \Phi(x)=\left(\begin{array}{c} x\left(t_{0}\right) \\ x^{\prime}\left(t_{0}\right) \\ \vdots \\ x^{(n-1)}\left(t_{0}\right) \end{array}\right) \quad \text { for each } x \in \mathrm{V} . $$ (a) Prove that \(\Phi\) is linear and its null space is the zero subspace of \(\mathrm{V}\). Deduce that \(\Phi\) is an isomorphism. Hint: Use Exercise 14 . (b) Prove the following: For any \(n\) th-order homogeneous linear differential equation with constant coefficients, any \(t_{0} \in R\), and any complex numbers \(c_{0}, c_{1}, \ldots, c_{n-1}\) (not necessarily distinct), there exists exactly one solution, \(x\), to the given differential equation such that \(x\left(t_{0}\right)=c_{0}\) and \(x^{(k)}\left(t_{0}\right)=c_{k}\) for $k=1,2, \ldots, n-1$.

Find the inverses of \(A=\left[\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 5 \\ 1 & 3 & 7\end{array}\right]\) and \(B=\left[\begin{array}{rrr}1 & -1 & 1 \\ 0 & 1 & -1 \\ 1 & 3 & -2\end{array}\right] .[\text { Hint: See Problem } 2.19 .]\)

Show (a) If \(A\) has a zero row, then \(A B\) has a zero row. (b) If \(B\) has a zero column, then \(A B\) has a zero column.

Suppose \(A\) and \(B\) are symmetric. Show that the following are also symmetric: (a) \(A+B\) (b) \(k A,\) for any scalar \(k\) \((\mathrm{c}) \quad A^{2}\) (d) \(A^{n},\) for \(n>0\) (e) \(f(A),\) for any polynomial \(f(x)\)
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