Problem

Determine an isomorphism between $\mathbb{R}^{2}$…

Problem 24 Hard Difficulty

Let $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ be a basis for the vector space $V,$ and suppose that $T_{1}: V \rightarrow V$ and $T_{2}: V \rightarrow V$ are the linear transformations satisfying
$$\begin{array}{ll}
T_{1}\left(\mathbf{v}_{1}\right)=\mathbf{v}_{1}+\mathbf{v}_{2}, & T_{1}\left(\mathbf{v}_{2}\right)=\mathbf{v}_{1}-\mathbf{v}_{2} \\
T_{2}\left(\mathbf{v}_{1}\right)=\frac{1}{2}\left(\mathbf{v}_{1}+\mathbf{v}_{2}\right), & T_{2}\left(\mathbf{v}_{2}\right)=\frac{1}{2}\left(\mathbf{v}_{1}-\mathbf{v}_{2}\right)
\end{array}$$
Find $\left(T_{1} T_{2}\right)(\mathbf{v})$ and $\left(T_{2} T_{1}\right)(\mathbf{v})$ for an arbitrary vector in $V$ and show that $T_{2}=T_{1}^{-1}$

Answer

$\left(T_{1} T_{2}\right)(\mathbf{v})=\left(T_{2} T_{1}\right)(\mathbf{v})=\mathbf{v}$

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Algebra

Differential Equations and Linear Algebra

Chapter 6

Linear Transformations

Section 4

Additional Properties of Linear Transformations

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Video Transcript

you wanted me to our basis off capital the and hope the one operates on both of them. You can write The Matrix was wanting to be one first wall of mist, even off even. And the coordinates of those are 11 and second calling will. But even off Lito, which is one comma minus one. Similarly, you can. I don't the Matrix course wanting toe to toe first William is one radio. One way toe and second column will be one Lehto minus one break. No, you want to find out the mattress? Corresponding toe dont too. And b 21. So let's multiply both of them out. So even off beautiful chorus want a product of the Obama matrices, which is equal. Do four sentries helpless off, which is one second entries, half minus off. Its zero curry entries have minus half, which is zero and 14 trees half place up, which is one. So the one of the two is the identity matrix. Now Vito off the one is you want to multiply the matrices to toe and viewing. So if you multiply them out, he will get the first entry is Harper's off, which is one second entry will be Let me write down them in the appropriate order. So you're Marty playing this with this? So the four sentries off myself. Second entries half minus office, zero Purdon trees. Zero for 10. Tree is one so teetered even also corresponds to identity. And from this you can see clearly that the matrix 42 is the inverse for matrix off. Even so, you too is actually called duty one in loose.

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