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3t t2 (5 points) Consider the matrix T1 ~t this matrix not invertible?For which values of t € R, if any; is5 points) Let T be a linear transformation such tha...

Question

3t t2 (5 points) Consider the matrix T1 ~t this matrix not invertible?For which values of t € R, if any; is5 points) Let T be a linear transformation such that T and T 3 2 Find a matrix representation of T. You may leave your answer as product of matrices.

3t t2 (5 points) Consider the matrix T1 ~t this matrix not invertible? For which values of t € R, if any; is 5 points) Let T be a linear transformation such that T and T 3 2 Find a matrix representation of T. You may leave your answer as product of matrices.



Answers

Assume that $T$ defines a linear transformation and use the given information to find the matrix of $T.$ $$\begin{aligned} &T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \text { such that } T(1,2,0)=(2,-1,1),\\ &T(0,1,1)=(3,-1,-1) \text { and } T(0,2,3)=(6,-5,4) \end{aligned}$$.

But we need to determine the nature trip visitation off our transformation. But the person we need to do is choose a basis for our domain on our coda mein. Since they're both are two, we're going to the simplest basis, which is the standard basis, which is 10 and 01 We now need to determine the image of the elements of our base other basis under the transformation. It's in this case t of 10 is going to give us three times one minus two times zero comma, one plus five times zero, which is equal to 31 And for his 01 we get is equal to three times zero minus two times one comma zero plus five times one which is equal to their to five. Now, we need to write the images of these trends of these basic of these basis elements as corner vectors in terms of the base under the basis off our coda name. In this case, we're gonna be writing them in terms of the standard basis, so 31 is gonna equal 30 plus zero one, which is equal three times 10 plus one times +01 and *** to five is equal to negative to zero plus zero five, which is equal to native to 10 Close 501 So now our corner vectors are going to be 31 and negative 25 And these corner vectors are going are going to be our column. Vectors for our matrix or matrix is going to be 31 Yeah, to five. So this is the Matrix representation off our linear transformation.

So to solve this problem we first knew choose a basis for the domain of transformation. Since the domain of our transformation is our three, we're going to choose a standard basis for three. But is it gonna be 100 comma 010 comma zero 01 Next, we need to compute the image of each of these elements of the basis. Under our transformation, she's gonna go something like this. T of 100 is equal to one plus five times zero minus three times zero is equal. One Teoh 010 is gonna equal zero plus five times one minus three times there, which is equal to five and t off. 00 one is gonna equal euro plus five times zero minor three times one, We're sequel minus three. So now we can compute the mixture representation of our linear transformation, which is going to be one by minus three

But to solve this problem first note that our Matrix A has three columns, which means we need to multiply our matrix A by a major three by one matrix of indeterminate variables, which is going to be X one next to X ray. And so this will give us two x one minus. Next to plus five times x three and three x one plus x two, plus three times back. Three. Translating this into Okay, then your transformation and this will give us TF x one Conlan TMX two gxf three is equal to two times except one minus x two. Was five times x three comma three times x one plus x two plus three times x ray.

Hello there. So for this exercise we have a little transformation defined by the matrix A. That is right here on one of the properties of our thorough matrices. Is that the the norm of the multiplication of a by some vector X. This is the norm of this multiplication will be equal to the norm of X. But in this case this matrix A. That defined this little transformation is a phenomenal and we have here Director X. So we need to verify that the norm of the leader transformation of X is equal to the norm of X. So let's just start by calculating what is the transformation of X to? The transformation of X is defined by the multiplication of this matrix A 4/5 0 minus 3/5 minus mine over 25. 4/5 minus 12/25. 12/25. 3/5. And the last one is 16/25 Times The Vector -235. So this multiplication give us the following A vector that is -23/5, 18/25 and 101 over 25. So the hard part here is to calculate the norm of this factor. So the norm of the linear transformation effects is equal to the norm of this vector, that it doesn't look quite easy to calculate Over 5, 18/25 and 100 pervert into five. And after you calculate these norm for this vector, you will obtain that. This is actually after me. All, the possible simplification equals the square root of 38. And if you take the the norm of the vector X. That is defined by the vector minus 235 You will see that it is actually discovered of 38. So Booth are the same and we have verified that it satisfied the conditions. Let's see.


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