Good evening Dr. Valerie, As an interested outsider I have also been busy, in addition to viewing your easy-to-follow Linear Algebra lectures playlist, with the first chapters of Otto Bretscher's Linear Algebra textbook, so I also looked at your video, if you don't mind. What I like about a question like number 5 is that it gives you a good indication of whether you have really understood the theory correctly! I have become much wiser the past few days thanks to you, and I'm still enjoying the process of learning more. Dr. Valerie, many thanks for your unwavering commitment to education!
For the projection of the matrix, would we not square root U1 squared plus U2 squared. I mean to say, for problem 1a, shouldn't it be 1/sqrt(17) and not 1/17?
1/17 is the correct multiplier here. I am using a formula to calculate the matrix. You can also project e1. We get 1/17 * (1,4) then project e2. We get 1/17 * (4,16). Then put these as columns in a matrix. The result is the same. I hope this helps.
Dear Dr. Valerie, I have a question regarding our reflection formula: 2 projL(vector x) - vector x . I was wondering why in the problem 1b (video 6:45), we could do 2 projL(vector x) - (the identity) ? shouldn't we minus the vector x?
Correct that reflection is 2projection(x)-x. If we think about this as a difference of functions, We have 2projection function minus the identity function (the identity maps every vector to itself). Thinking this way (as a difference of functions), we can translate to the corresponding matrices. The 2x2 matrix of reflection is therefore 2projection matrix minus identity matrix. I hope this helps.
Probably this summer I want to make a website with all my worksheets/reviews/etc. Until then, here is this worksheet: drive.google.com/file/d/1fLCl3ue76TPHzFBGjzcC0xEJwBqT3_Or/view?usp=sharing
Generally one can solve a linear system at REF. However to find the inverse of a 3x3 matrix, we are really solving 3 linear systems: Ax=e1, Ax=e2, and Ax=e3. I think it is more work to stop at REF back substitute three times.
In our textbook, determinant comes in chapter 6. Inverse of a matrix is in chapter 2. The determinant itself can be a complicated calculation. Is there a formula for the inverse that uses the determinant? Probably. There is certainly a shortcut for 2x2 matrices using it. But, I don't see how that method could be generally shorter than the standard one used here.
Good evening Dr. Valerie, As an interested outsider I have also been busy, in addition to viewing your easy-to-follow Linear Algebra lectures playlist, with the first chapters of Otto Bretscher's Linear Algebra textbook, so I also looked at your video, if you don't mind. What I like about a question like number 5 is that it gives you a good indication of whether you have really understood the theory correctly! I have become much wiser the past few days thanks to you, and I'm still enjoying the process of learning more. Dr. Valerie, many thanks for your unwavering commitment to education!
You are welcome :). I appreciate your kind words.
For the projection of the matrix, would we not square root U1 squared plus U2 squared. I mean to say, for problem 1a, shouldn't it be 1/sqrt(17) and not 1/17?
1/17 is the correct multiplier here. I am using a formula to calculate the matrix. You can also project e1. We get 1/17 * (1,4) then project e2. We get 1/17 * (4,16). Then put these as columns in a matrix. The result is the same. I hope this helps.
Dear Dr. Valerie, I have a question regarding our reflection formula: 2 projL(vector x) - vector x . I was wondering why in the problem 1b (video 6:45), we could do 2 projL(vector x) - (the identity) ? shouldn't we minus the vector x?
Correct that reflection is 2projection(x)-x. If we think about this as a difference of functions, We have 2projection function minus the identity function (the identity maps every vector to itself). Thinking this way (as a difference of functions), we can translate to the corresponding matrices. The 2x2 matrix of reflection is therefore 2projection matrix minus identity matrix. I hope this helps.
It's very helpful. Thank you very much for your time!
Can you upload the practice problems please.
Probably this summer I want to make a website with all my worksheets/reviews/etc. Until then, here is this worksheet: drive.google.com/file/d/1fLCl3ue76TPHzFBGjzcC0xEJwBqT3_Or/view?usp=sharing
Is there another way to find the inverse without using rref?
Generally one can solve a linear system at REF. However to find the inverse of a 3x3 matrix, we are really solving 3 linear systems: Ax=e1, Ax=e2, and Ax=e3. I think it is more work to stop at REF back substitute three times.
@@DrValerieHower ok its because I saw a different video where they solve a 3x3 matrix using matrix of minors and cofactors, adjugate and determinant.
In our textbook, determinant comes in chapter 6. Inverse of a matrix is in chapter 2. The determinant itself can be a complicated calculation. Is there a formula for the inverse that uses the determinant? Probably. There is certainly a shortcut for 2x2 matrices using it. But, I don't see how that method could be generally shorter than the standard one used here.