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Repeat Exercise $35,$ assuming $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors of $A$ that correspond to eigenvalues $-1$ and $3,$ respectively.

As in Exercise 35, T (u) = ?u and T (v) = 3v because u and v are eigenvectors for the eigenvalues ?1 and 3, respectively, of the standard matrix A. Again, because T is linear, the image of w is T(w) = T(u + v) = T(u) + T(v).

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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her case are on this graph. Here we have two vectors. Are you on fee? Are we going to define w to be you? Plus fee? So particularly, it's going to be this effect. The hell Okay, on what we going to do is defined a linear mapping t of X equal to a Times X. We're going to shoot that u and V. I can. Victor's off the matrix A which corresponds to the Eiken values minus one and three, respectively. Okay, so what we going to do is we're going to be graphing. Ah, tee off year T O V and T off w. Okay. All right. So let's start with the 1st 2 t of you and t a fee now because, uh, you and via Aiken vectors of a Then we must have that eight times X equal to land the times X for the particular Aiken vectors for the particular eye and values that we have. So because you is in Aiken Vector off, eh? For the Eiken value. Modest one t of you is then just minus you. So if I do it and blue here, that would be tee off. You okay for the same reason T V should then be three times the vector fee. So the vet t a V should be Now. It's probably hard to get off the page, but all this paper. But it's there. Yes, it is. A shoot V is smaller. S 05 rub Oldest Celts on a spit bat. But that dumb not too fast. Let's just say fee is there. We got the arrays of still day. Okay, so that's Fi. So then three V's probably say there they will call this TV okay for T A W. So because, uh, the mapping T is linear. This has to be tee off year plus T off V. So it's just the vector. Some off these two things. It's a It's a bit hard to do it on the computer, but I swear if you have a rule of it makes it 10 times easier. But it's gonna be that effective there. This is going to be t ough W, which is defective. Some off t of you on TV away

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