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In Exercises $27-30$ , use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.$$(1-t)^{2}, t-2 t^{2}+t^{3},(1-t)^{3}$$

$(1-t)^{2}, t-2 t^{2}+t^{3},(1-t)^{3}$ are not linearly independent.

Calculus 3

Chapter 4

Vector Spaces

Section 4

Coordinate Systems

Vectors

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okay. We have to test the linear independence of this sort of polynomial. And the questions that are given is one minus T Squire on D minus to D squared plus D u and one minus T to leave our whole cube. And as we simplify that, we will. We can write this one minus two p plus t square. And this is simplified already. So t minus two t square plus teach you by simplifying that we can write it one minus three D plus three d Squire minus D cube. Okay, now we can. Alright, directors off these equations. These are Tzeitel. It's coefficient off. Did you? This is coefficient off the square. This is coefficient off T. And this is constant. So same here. It will be one minus two, one and zero. And in this question, it will be minus 13 my industry and one. And if we have to check the linear dependence independency, we have to put these metrics in these vectors. In this equation, he x equals to zero. And if zero is the only solution, the vectors are linearly dependent. So now we will put the values in the metrics. It will be zero one minus 21 one minus 210 minus 13 minus three and one is one Here and the solution will be zero Zito, Zito! Zito Okay, now we all this all this First we will do, um in rotary we will go to our two plus r three and in drove for we will do minus r one plus r four So we will get Zito one minus one Luisito one My next to Titi Zito Zito Monastery Titty Zito A Tzeitel Tau minus 20 Okay, now again, we have to solve it again And we will do in r three r three equals two Terry R one plus r three and in our four they will do minus two r one plus are four and we will get zero Who won my next one Brazil one minus 230 Zito Zito Zito Zito 00 Zito Zito Okay, so in this equation from the road three and RAV four, we can say that. Okay, First, it can say that X two minus X three equals to zero. So next week was toe x three Because there will be so many way lose off extra hair and from the raw, too from the road to means this means this role. We can say x one minus two x two plus three x three equals to zero. That means Exxon. It was 22 x two minus three X three. Okay, that will be again. I will put the value two x two s two x three minus three x three equals two minus x three So finally, finally X t is very variable. Ex tree is free Variable. So so the sad is not linearly independent. This is all fine alongside. Thank you.

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