The Inverse and the Transpose
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0:46Recap our current situation
0:46Recap our current situation
0:46Recap our current situation
1:45Start thinking about how we'll transform Normals
1:45Start thinking about how we'll transform Normals
1:45Start thinking about how we'll transform Normals
3:51Blackboard: Rotating Normals (is pretty straightforward)
3:51Blackboard: Rotating Normals (is pretty straightforward)
3:51Blackboard: Rotating Normals (is pretty straightforward)
5:01Blackboard: Non-uniform scaling gets a little bit hairy
5:01Blackboard: Non-uniform scaling gets a little bit hairy
5:01Blackboard: Non-uniform scaling gets a little bit hairy
6:17Blackboard: Pretending we have the edge of a gem
6:17Blackboard: Pretending we have the edge of a gem
6:17Blackboard: Pretending we have the edge of a gem
8:41Blackboard: Vectors are not all the same
8:41Blackboard: Vectors are not all the same
8:41Blackboard: Vectors are not all the same
11:23Blackboard: Normals are written differently in linear algebra
11:23Blackboard: Normals are written differently in linear algebra
11:23Blackboard: Normals are written differently in linear algebra
12:46Blackboard: Sometimes you have to go down a math holeα
12:46Blackboard: Sometimes you have to go down a math holeα
12:46Blackboard: Sometimes you have to go down a math holeα
13:03Blackboard: Constructing the P vectors
13:03Blackboard: Constructing the P vectors
13:03Blackboard: Constructing the P vectors
14:17Blackboard: Matrix multiplication
14:17Blackboard: Matrix multiplication
14:17Blackboard: Matrix multiplication
24:18Blackboard: Transpose operation
24:18Blackboard: Transpose operation
24:18Blackboard: Transpose operation
26:19Blackboard: Inverse operation
26:19Blackboard: Inverse operation
26:19Blackboard: Inverse operation
30:24Blackboard: Gauss steps in with Gaussian Elimination
30:24Blackboard: Gauss steps in with Gaussian Elimination
30:24Blackboard: Gauss steps in with Gaussian Elimination
35:26Blackboard: Solving equations in multiple unknowns
35:26Blackboard: Solving equations in multiple unknowns
35:26Blackboard: Solving equations in multiple unknowns
36:33Blackboard: We're just entirely in the math holeβ
36:33Blackboard: We're just entirely in the math holeβ
36:33Blackboard: We're just entirely in the math holeβ
39:02Blackboard: We can regularise operations on the rows and columns of this matrix
39:02Blackboard: We can regularise operations on the rows and columns of this matrix
39:02Blackboard: We can regularise operations on the rows and columns of this matrix
39:49Blackboard: Use numbers to clearly demonstrate Gaussian Elimination
39:49Blackboard: Use numbers to clearly demonstrate Gaussian Elimination
39:49Blackboard: Use numbers to clearly demonstrate Gaussian Elimination
41:12Blackboard: This is not JZ's termγ
41:12Blackboard: This is not JZ's termγ
41:12Blackboard: This is not JZ's termγ
41:27Blackboard: You can then divide the last remaining term by whatever the target is
41:27Blackboard: You can then divide the last remaining term by whatever the target is
41:27Blackboard: You can then divide the last remaining term by whatever the target is
43:10Blackboard: We want to be able to multiply a matrix by something in order to produce that identity matrix
43:10Blackboard: We want to be able to multiply a matrix by something in order to produce that identity matrix
43:10Blackboard: We want to be able to multiply a matrix by something in order to produce that identity matrix
44:38Blackboard: The regular solution form for Gaussian Elimination
44:38Blackboard: The regular solution form for Gaussian Elimination
44:38Blackboard: The regular solution form for Gaussian Elimination
58:12Blackboard: Unfortunately we didn't quite get to the inverse transpose
58:12Blackboard: Unfortunately we didn't quite get to the inverse transpose
58:12Blackboard: Unfortunately we didn't quite get to the inverse transpose
1:02:26Q&Aδ
1:02:26Q&Aδ
1:02:26Q&Aδ
1:03:06 For inverting 2x2 matrices, a simple cofactor equation is quite efficient
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1:03:06 For inverting 2x2 matrices, a simple cofactor equation is quite efficient
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1:03:06 For inverting 2x2 matrices, a simple cofactor equation is quite efficient
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1:03:26 All of the steps in elimination can be represented as a matrix. Starting with M and multiply by all those matrices you get the identity. So those matrices multiplied together are the inverse. Multiplying them all by the identity gives you the inverse. Is that the trick you are alluding to?
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1:03:26 All of the steps in elimination can be represented as a matrix. Starting with M and multiply by all those matrices you get the identity. So those matrices multiplied together are the inverse. Multiplying them all by the identity gives you the inverse. Is that the trick you are alluding to?
🗪
1:03:26 All of the steps in elimination can be represented as a matrix. Starting with M and multiply by all those matrices you get the identity. So those matrices multiplied together are the inverse. Multiplying them all by the identity gives you the inverse. Is that the trick you are alluding to?
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1:07:47 To find the inverse matrix couldn't we just rotate by the negative angle and scale by 1 over the amount that we scaled by originally?
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1:07:47 To find the inverse matrix couldn't we just rotate by the negative angle and scale by 1 over the amount that we scaled by originally?
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1:07:47 To find the inverse matrix couldn't we just rotate by the negative angle and scale by 1 over the amount that we scaled by originally?
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1:17:32 I remember doing this by putting the identity besides the original matrix and performing each step to each
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1:17:32 I remember doing this by putting the identity besides the original matrix and performing each step to each
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1:17:32 I remember doing this by putting the identity besides the original matrix and performing each step to each
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1:21:24 If you had a matrix such as int[2][2] with the example of abcd as the variables in place, couldn't you just swap a and d and make c and b negative?
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1:21:24 If you had a matrix such as int[2][2] with the example of abcd as the variables in place, couldn't you just swap a and d and make c and b negative?
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1:21:24 If you had a matrix such as int[2][2] with the example of abcd as the variables in place, couldn't you just swap a and d and make c and b negative?
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1:22:31 I understood none of today's episode. Would you recommend that I rewatch it, or do you think all may become clear tomorrow?
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1:22:31 I understood none of today's episode. Would you recommend that I rewatch it, or do you think all may become clear tomorrow?
🗪
1:22:31 I understood none of today's episode. Would you recommend that I rewatch it, or do you think all may become clear tomorrow?
🗪
1:23:31 If you write A and then I next to each other, and apply a bunch of operations, you are computing T3T2T1 A and T3T2T1 I. Because you ended up at the identity, T3T2T1A = I. So T3T2T1 = A^-1. Therefore T3T2T1 I = A^-1
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1:23:31 If you write A and then I next to each other, and apply a bunch of operations, you are computing T3T2T1 A and T3T2T1 I. Because you ended up at the identity, T3T2T1A = I. So T3T2T1 = A^-1. Therefore T3T2T1 I = A^-1
🗪
1:23:31 If you write A and then I next to each other, and apply a bunch of operations, you are computing T3T2T1 A and T3T2T1 I. Because you ended up at the identity, T3T2T1A = I. So T3T2T1 = A^-1. Therefore T3T2T1 I = A^-1
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1:26:46Blackboard: Look at all of this maths
1:26:46Blackboard: Look at all of this maths
1:26:46Blackboard: Look at all of this maths
1:27:30Blackboard: Setup for Day 102 and sign off
1:27:30Blackboard: Setup for Day 102 and sign off
1:27:30Blackboard: Setup for Day 102 and sign off