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3 . Find the standard matrix of the linear transformation T: Ri through the plane T1 T2, and find the image of vector (1,2,3).R' that reflects each vector...

Question

3 . Find the standard matrix of the linear transformation T: Ri through the plane T1 T2, and find the image of vector (1,2,3).R' that reflects each vector

3 . Find the standard matrix of the linear transformation T: Ri through the plane T1 T2, and find the image of vector (1,2,3). R' that reflects each vector



Answers

The images of the standard basis vectors for $R^{3}$ are given for a linear transformation $T: R^{3} \rightarrow R^{3}$. Find the standard matrix for the transformation, and find $T(x).$ $$T\left(\mathbf{e}_{1}\right)=\left[\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right], T\left(\mathbf{e}_{2}\right)=\left[\begin{array}{r} -3 \\ -1 \\ 0 \end{array}\right], T\left(\mathbf{e}_{3}\right)=\left[\begin{array}{l} 1 \\ 0 \\ 2 \end{array}\right] ; \quad \mathbf{x}=\left[\begin{array}{l} 3 \\ 2 \\ 1 \end{array}\right]$$

Right. The key idea for this problem is to get the standard matrix for T. If you know what he does too, The base vectors E one, E two and E three. All you have to do is make the matrix A. Where T one is the first column. T E two is the second column, and T E three is the third column. So, for example, if you know t of E one is 130 and you know T of each you Is the Vector 001. And you know, th e three Is the vector four -3 -1. Then you can get the standard matrix for tea. Just by listing these guys as the columns. Okay? So A Is going to equal 130 001 four -3 -1. Okay. And that might make some sense to you guys because if you were going to do a times E one and again, if you remember E one is the vector 100 then you're gonna just pick off The 130 when you do that matrix multiplication. So again here is my standard matrix. And then if I need to compute T. Of some other vector T. Of X. And in this example X. Is equal to The Vector 2 10. All I have to do to figure out what T. Is is just multiply A times X. So that matrix 130001 4 -3 -1. Going to multiply that by 210. And when we do this multiplication I get one times two plus zero times one plus four times zero. I get three times two plus zero times one minus three times zero. Don't forget how you do your matrix multiplication zero times two Plus one times 1 -1 times zero. Mhm. And when I simplify all of that, I get the vector to for the first component, six for the second component and one for the third component. Awesome. All right, these ones are pretty easy. Try a few more. Have a good day.


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