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Let $A=\left[\begin{array}{ccc}{5} & {0} & {0} \\ {0} & {.5} & {0} \\ {0} & {0} & {.5}\end{array}\right], \mathbf{u}=\left[\begin{array}{r}{1} \\ {0} \\ {-4}\end{array}\right],$ and $\mathbf{v}=\left[\begin{array}{c}{a} \\ {b} \\ {c}\end{array}\right]$
Define $T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by $\vec{T}(\mathbf{x})=A \mathbf{x} .$ Find $T(\mathbf{u})$ and $T(\mathbf{v})$

In this question we need to find in the value of transport of a place to point Tracy. So first of all I am going to put diameter tricks and then we need to add 2.3 times of the see matrix. And then we have to make that transport. So I'm going to put and see metrics. So a my taxi is having Elements 1.5 -2.35 And then 5.6 44.2, zero, 12.2 and see my tricks is having 10 2030 and minus 10 -20 then -30. Now you can see that. First of all we have to make The 2.3 times of the sea metrics and then we can add both of the metrics. So I'm going to make 2.3 times of age to the age of the element. So we can see that We are going to multiply 2.3 which each of the elements. So initially we get 23 Then if you multiply with 20 We get 46. And then if we multiply with 30 We got 69 then similarly -23, 4 -46 -69. And after adding both of them, we need to make their transport. No we have to add the corresponding elements. Then we have to make the transport means we have to add 1.5 with 23 because these are the Past elements. So we have got a corresponding elements. So we get 24.5, no ed -2.35 with 46. So we get 43 points 65. Then 5.6 with 16 9. So we get 74.6 And then 44.2 with minus 23. So we get 21.2 And then zero with the -46. So we get -46 and then 12.2 With -69. So we get -56.8. Now after making their transport we know that we have to change roll into column and column in turo. So made the first row first column 24 blind flight When they want went to side made the 1st Parliament to first of all. Similarly second column into second Will 43.65 -46 74.6 -56.8. Okay so this is no we're law of the given question here and that was a plus 2.3 C. And it's transposed. OK thank you.

Okay. We are given two vectors and we're asked to find the derivative of there dot product and the derivative of their cross products. So let's start with um just the dot product. And then we'll take its derivative. So we multiply the eye components. We add that to the multiple of the J. And add that to the multiple K. Remembering that this is Scaler. And so we don't keep the IGs in case we just are using the values in them. So now to take the derivative, I will take the derivative of that result, which is 12 T. to the third kate. Now for our cross product. Um let's go ahead and write out those I. J. S. And cays. And so we can use our kind of matrix. See here to um find that cross product. Okay, so first of all um our first values are going to come from if you hide column I write you do a criss cross and um and we'll take that negative T squared times T. The third. And we get negative T. To the fifth and then we take the negative T. Squared um times T. So we'll do a subtract. But then you can see that is a negative so it ends up plus. So that is our eye piece. Now we're gonna hide RJ. And so we get the T. To the sixth minus T. Squared and then finally hide the K. And you get that negative T. To the fifth minus a negative. So that will become plus T. To third. Okay so now we're just taking the derivative of this cross product. So when we do that we have a negative five T. To the fourth plus three T squared. That's all in our eye direction. And then we have a six T. To the fifth minus two T. In the J. Direction. And then a negative five T. To the fourth Plus three T. Squared in the K. direction.

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In this question we have to find The transport of difference of 2.1 A And 2.3 c. So first of all we need to put the A matrix and and then -2.3. See metrics here and after all we are to make that transpose it. So I'm going to substitute a metrics and see metrics. So he's having elements 1.5 -2.35 5.6 and 44.2 0 12.2. And steve's having 10 2030 -10 -20 -30. So we can see here. No we have to make 2.1 times of age of the element. So I'm going to multiply Each element with 2.1. So first of all, 1.5 is multiplied with 2.1. So we got 3.15. Then we have to multiply -2.35. So we get -4 935 And then 5.6 is there? So let's see You guarantee our 11.76 Then 44.2 is there? So we get here 92.82 and Then we can see we have zero so we get zero then we have 12.2. So we get here 25.62. After that we have to make Product of 2.3 which each of the element. So we've got 23 46 69 -23 -46 -169. And finally at last we have to make that transpose er Okay so see now we have to subtract so I'm going to subtract here and then we will make their transport. So subtract three 15 and 23. So we get -19 points 85. Then subtract minus four point 935 and 46. So we get minus 50 .935 and then 11.76 and 69. So we get -57.24, Then 92 82 and minus 23. So you can see that We get 115.8-10 0 and -46. So we get 46 And then 25.62 and 69. So we get here 94.62. Now we have to make finally that transport, it means that we have to change the first column into first of all, So -19.85 115.82 and minus 50 point wait a while I am going to relight it. I need more space so I'm providing space Here. So first of all -19.85, 115.82, second of all minus 50.935 46 3rd level minus 57.24 and 94.62. Okay, this is the answer for the given question here, Thank you