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Chapter 2: Problem 14

Prove the converse of Exercise 8: If \(A\) and \(B\) are each \(m \times n\) matrices with entries from a field \(F\), and if there exist invertible $m \times m\( and \)n \times n\( matrices \)P\( and \)Q$, respectively, such that \(B=P^{-1} A Q\), then there exist an \(n\)-dimensional vector space \(\mathrm{V}\) and an \(m\)-dimensional vector space \(\mathrm{W}\) (both over \(F\) ), ordered bases \(\beta\) and \(\beta^{\prime}\) for \(\mathbf{V}\) and \(\gamma\) and \(\gamma^{\prime}\) for \(\mathbf{W}\), and a linear transformation $\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}$ such that $$ A=[\mathrm{T}]_{\beta}^{\gamma} \text { and } B=[\mathrm{T}]_{\beta^{\prime}}^{\gamma^{\prime}} . $$ Hints: Let $\mathrm{V}=\mathrm{F}^{n}, \mathrm{~W}=\mathrm{F}^{m}, \mathrm{~T}=\mathrm{L}_{A}\(, and \)\beta\( and \)\gamma$ be the standard ordered bases for \(\mathrm{F}^{n}\) and \(\mathrm{F}^{m}\), respectively. Now apply the results of Exercise 13 to obtain ordered bases \(\beta^{\prime}\) and \(\gamma^{\prime}\) from \(\beta\) and \(\gamma\) via \(Q\) and \(P\), respectively.

Short Answer

Expert verified
In summary, let V = F鈦� and W = F岬� be the n-dimensional and m-dimensional vector spaces over field F, respectively, and T: V 鈫� W be the linear transformation defined by T = L鈧� (mapping matrix multiplication by A). Now, by applying the results of Exercise 13 to the given conditions for matrices A and B, we have established that there exist invertible matrices P and Q such that B = P^{-1}AQ. Then, we derived new ordered bases 尾' and 纬' by multiplying 尾 and 纬 with matrices Q and P, respectively. Finally, we proved that A = [T]鈧嶀郸;岬�, and B = [T]鈧嶀郸';岬�', establishing the desired representation for A and B with respect to the different bases.

Step by step solution

01

Define vector spaces, linear transformation, and bases

Let vector spaces V = Fn and W = Fm. Define the linear transformation T: V 鈫� W as T = LA, mapping matrix multiplication by A. Also, let 尾 and 纬 be the standard ordered bases for V and W, respectively.
02

Recall the results of Exercise 13

Exercise 13 states that for a linear transformation of vector spaces over a field F, and given ordered bases 尾 and 纬, there exist invertible matrices P and Q such that: \[B = P^{-1}AQ\]
03

Apply Exercise 13 to our problem

We will apply Exercise 13 to the given conditions for matrices A and B and the linear transformation T. There exist invertible matrices P and Q such that: \[B = P^{-1}AQ\]
04

Obtain the new bases from 尾 and 纬 using P and Q

We will obtain the new ordered bases 尾' and 纬' by performing the following operations: 1. For 尾': Multiply each vector in 尾 by matrix Q to get 尾'. 2. For 纬': Multiply each vector in 纬 by matrix P to get 纬'.
05

Prove the required representation of A and B

Now we prove that A and B can be represented by T with respect to the different bases: 1. A = [T]尾^纬: Since T = LA, we already have A in the desired form with respect to ordered bases 尾 and 纬. 2. B = [T]尾'^纬': From Step 3, we know that B = P^(-1)AQ. Using the new ordered bases 尾' and 纬', we have: \[ [\mathrm T]_{\beta^{\prime}}^{\gamma^{\prime}} = P^{-1} [\mathrm T]_{\beta}^{\gamma} Q = P^{-1} AQ \] Comparing with the result from Step 3, we can see that B = [T]尾'^纬'. Thus, we have proven that A=[T]尾^纬 and B=[T]尾'^纬' satisfying the required condition.

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Most popular questions from this chapter

For each of the following parts, determine whether the statement is true or false. Justify your claim with either a proof or a counterexample, whichever is appropriate. (a) Any finite-dimensional subspace of \(C^{\infty}\) is the solution space of a homogeneous linear differential equation with constant coefficients. (b) There exists a homogeneous linear differential equation with constant coefficients whose solution space has the basis \(\left\\{t, t^{2}\right\\}\). (c) For any homogeneous linear differential equation with constant coefficients, if \(x\) is a solution to the equation, so is its derivative \(x^{\prime}\). Given two polynomials \(p(t)\) and \(q(t)\) in \(\mathrm{P}(C)\), if $x \in \mathrm{N}(p(\mathrm{D}))\( and \)y \in\( \)\mathrm{N}(q(\mathrm{D}))$, then (d) \(x+y \in \mathrm{N}(p(\mathrm{D}) q(\mathrm{D}))\). (e) \(x y \in \mathrm{N}(p(\mathrm{D}) q(\mathrm{D}))\).

Find a basis for each of the following subspaces of \(C^{\infty}\). (a) \(N\left(D^{2}-D-1\right)\) (b) \(N\left(D^{3}-3 D^{2}+3 D-1\right)\) (c) \(\mathrm{N}\left(\mathrm{D}^{3}+6 \mathrm{D}^{2}+8 \mathrm{D}\right)\)

Label the following statements as true or false. Assume that \(V\) and \(\mathrm{W}\) are finite-dimensional vector spaces with ordered bases \(\beta\) and \(\gamma\), respectively, and $\mathrm{T}, \mathrm{U}: \mathrm{V} \rightarrow \mathrm{W}$ are linear transformations. (a) For any scalar \(a, a \mathbf{T}+\mathbf{U}\) is a linear transformation from \(\mathbf{V}\) to \(\mathbf{W}\). (b) \([\mathrm{T}]_{\beta}^{\gamma}=[\mathrm{U}]_{\beta}^{\gamma}\) implies that \(\mathrm{T}=\mathrm{U}\). (c) If \(m=\operatorname{dim}(\mathrm{V})\) and \(n=\operatorname{dim}(\mathrm{W})\), then \([\mathrm{T}]_{\beta}^{\gamma}\) is an \(m \times n\) matrix. (d) \([\mathrm{T}+\mathrm{U}]_{\beta}^{\gamma}=[\mathrm{T}]_{\beta}^{\gamma}+[\mathrm{U}]_{\beta}^{\gamma}\). (e) \(\mathcal{L}(\mathrm{V}, \mathrm{W})\) is a vector space. (f) \(\mathcal{L}(\mathrm{V}, \mathrm{W})=\mathcal{L}(\mathrm{W}, \mathrm{V})\).

Let \(A\) and \(B\) be \(n \times n\) matrices such that \(A B\) is invertible. (a) Prove that \(A\) and \(B\) are invertible. Hint: See Exercise 12 of Section 2.3. (b) Give an example to show that a product of nonsquare matrices can be invertible even though the factors, by definition, are not.

Pendular Motion. It is well known that the motion of a pendulum is approximated by the differential equation $$ \theta^{\prime \prime}+\frac{g}{l} \theta=0, $$where \(\theta(t)\) is the angle in radians that the pendulum makes with a vertical line at time \(t\) (see Figure 2.8), interpreted so that \(\theta\) is positive if the pendulum is to the right and negative if the pendulum is to the left of the vertical line as viewed by the reader. Here \(l\) is the length of the pendulum and \(g\) is the magnitude of acceleration due to gravity. The variable \(t\) and constants \(l\) and \(g\) must be in compatible units (e.g., \(t\) in seconds, \(l\) in meters, and \(g\) in meters per second per second). (a) Express an arbitrary solution to this equation as a linear combination of two real-valued solutions. (b) Find the unique solution to the equation that satisfies the conditions $$ \theta(0)=\theta_{0}>0 \text { and } \theta^{\prime}(0)=0 \text {. } $$ (The significance of these conditions is that at time \(t=0\) the pendulum is released from a position displaced from the vertical by \(\theta_{0}\).) (c) Prove that it takes \(2 \pi \sqrt{l / g}\) units of time for the pendulum to make one circuit back and forth. (This time is called the period of the pendulum.)
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