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In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.$$A=\left[\begin{array}{rrr}{4} & {0} & {1} \\ {-2} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right], \lambda=1,2,3$$

$\left\{\left[\begin{array}{c}{0} \\ {1} \\ {0}\end{array}\right],\left[\begin{array}{c}{-1} \\ {2} \\ {2}\end{array}\right],\left[\begin{array}{c}{-1} \\ {1} \\ {1}\end{array}\right]\right\}$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

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So we want to find basis for Reagan spaces Forgiven values off lander. Right? So just street off the bed. Let's scale clip. Let's stay agree to be extra next to extreme And for any Lambda just lets us calculate what these what? This equality gives us the conditions for extra, extra extra. So our is or one on our last rule is is x one x two x three on this is Lambda explode Lived a extra No, no XLT. Okay. No, The first equation is just four times x one less x tree. It calls Landra him sex room. So this gives us that extra is a quartile Lambda minus four games X one. Okay, second equation is negative. Two times X one close extraordinaire equals under dames. Extreme. This gives us negative door dames. Exxon, is it quartile? Lambda minus one times extract. Oh, okay. And the third equation is negative. Two times x one. My 63 equals lander times extreme. This gives us negative two times. X one is a quarto lambda minus one time 63 So combining these two equations combining the 2nd and 3rd equation. You can see that. Okay, so let's plug in specific values off land and see what we get. So when Lam Days equaled one first equation tends you that extreme sick weirdo Negative three off excellent and minus two. Excellent is zero. So that gives you X one is zero and tired. Equation also tells you that. Excellent. Zero. Okay, so if excellent zero the next 30 Okay, on extra can be anything. So you're Agon. Victor. Look like zero extra deal. So one basis basis, which consists off. One element for this is, uh, Zito. Okay, so that's your case for Lambda equals one. So the second cases when Lambda is equal, Toto. Okay, so when lambda is in Quito this go back to our equations. First equation tells you that X three is equal to negative toe affects. Want if you look closely at Equation three that tells you that negative to off X one is equal to x tree, which is basically our first equation. OK, so 1st and 3rd equation gives you the same thing. And then the second equation is that negative bull affects one has a little extra. So you're Agon. Victor's look like Well, you have extra and the next reason called Works too. And then Excellent. Is it, Gordo? Negative. Extra divided point. Okay, so one basis would be like basis. That consist off. One element can be negative. One over to one and one. So a third and final cases when Landry's accord with three. So when land is equal to three, just look back at our equations. So the first equation is simply XT resicore do negative off x one. Uh, negative to off X one is a Kordell two off extra and third equation is negative. Two off X one is a core do two off extreme with the same as the first equation. And the second equation tells you that negative. Excellent. His acquittal fixed. Okay, so here is your excellent extra sick or the negative affects on an extra Z quarter. Negative. So basis is a single victor. And you can take this Toby one negative one. Okay,

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