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Question 1 (5 points) . Let T Rn Rm be a linear transformation; i.e. T(21, A is an by n matrix. Calculate Jacobian of T.En )Ax whereQuestion 2 (10 points) . Let f(...

Question

Question 1 (5 points) . Let T Rn Rm be a linear transformation; i.e. T(21, A is an by n matrix. Calculate Jacobian of T.En )Ax whereQuestion 2 (10 points) . Let f(€,y) 23 +y2_ Show that (0,0) is a saddle point . Note that you cannot use the second derivative test for this function_ Hint: Find the curve of intersection of the graph of with the :z-plane _

Question 1 (5 points) . Let T Rn Rm be a linear transformation; i.e. T(21, A is an by n matrix. Calculate Jacobian of T. En ) Ax where Question 2 (10 points) . Let f(€,y) 23 +y2_ Show that (0,0) is a saddle point . Note that you cannot use the second derivative test for this function_ Hint: Find the curve of intersection of the graph of with the :z-plane _



Answers

$1-6$ Find the Jacobian of the transformation.
$$
x=u^{2}+u v, \quad y=u v^{2}
$$

1 20 year Couldn't assault Cover number six eggs sequence repress W squared. Why equals w plus you square? Is that because you plus the square So the Corbyn metrics is given by So at X y zed that respect toe you VW, which is because 01 Ow two u 011 to the little So which comes to be, like zero in do zero into zero minus one named a totally minus one Indu do you into zero minus one into one plus do the blue window to you in tow to remind us zero in the one It comes to be one plus eight u v w that sent off a question

So we want to find a two by two matrix that reflects points through the horizontal access first and then reflects through Ah, the line X two equals x one. So the first rotation will call the Are one is tthe e reflection through the horizontal X one axis. Ah, so if you look at page, I believe that 74 in the section and it gives all of the EU's general rotations and the rotation for this ISS 10 zero negative one because it slips the why coordinates. So next, the second rotation we need to reflect through the line X two equals equals one. Um, so if you think about this actually happening, um, that line is right here. So if you pick a point, let's say this one here and you reflect over this line the X coordinate in the Y coordinate are switching. So, um, or you could just look on patient before the rotation matrix for this is 0110 So we're switching the X on my coordinates. So the combination of the reflections is tthe e ah. Multiplication of these major cities with the 1st 1 has to be on the end because that's the Matrix we'd would apply first. And then we would apply The Second Matrix are too. So if we multiply these two things together, let's see what we get. So our first, uh, coordinate is zero plus zero, which is your own. Next we take second call. Sometimes the first row, so zero plus minus. One is minus one. Next second Road times first call home. One plus zero is zero. First, I want 01 And lastly, second row time, Second column. We get zero plus zero, which is zero. So this is our final, um, no tricks.

Uh huh. In the current problem, um, we had given a situation the scene, if two different transformations actually doing the same work or not. First, they're talking about the transformation that reflects a point through. Excellent. And then next. Okay, so we will see. What would there transformation look like? So take even and e to. So if we are reflecting through X one axis one common zero. Okay, then this is not reflecting anywhere because with respect to X, when access there's the middle most positions, there is no change. So this becomes 10 where it is this point Over here, this comes down over here, that is zero former minus, right? No. De So if there is a reflection X when we know this is the standard metrics now they're telling will Now you change this How reflection Through extra. So if there is a reflection toe extreme Now we forget about this point we're having now only this point and this point. So this will change to hear. Whereas this will remain constant. So that's why we will get minus one. We'll see toe zero minus one. So this is our final step metrics for this transformation. This is our teeth. No. They're telling that show that he can also be described as a linear transformation that rotates points about the auditing. And what is the angle of the rotation now? See, initially, point was here East, this is the origin. And to be rooted about the or eating so from the origin, whatever distances here, the same distance that you travel. Headache. So what What are we finding is this is the angle we've been here. This is the anger. Okay, So when we I want to find the metrics of transformation, we know that the metrics of transformation through the origin they are rotation through the origin. We know that metrics that standard metrics is off the form minus one zero zero minus one. Right. Which means what if you remember, there is a photograph of a fish over here and that real. So this will come here. This will go here. That's the idea. Reflection are rotating through the origin. So now if you see both the transformation metrics a scene, hence we can see that both, um, addresses, uh, signifying the same transformation. Hence reflecting through first X one and then x two all rotating about the origin is actually the same action. That's something


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