Open course ware

ANA RITA PIRES: Hi. Welcome back to recitation. In lecture, you've been learning about the properties of determinants. To remember, there were three main properties, and then seven more that fall out of those three. I'll tell you what these three were. The first one was the determinant of the identity matrix is always equal to 1. If you switch two rows in a matrix, the determinant switches sign. And the determinant is a function of each-- it's a linear function of each row separately. And there's seven more. We'll use them here. Today's problem is about finding the determinants of matrices by using these properties. So here you have four matrices. A has lots of 100's, 200's, and 300's numbers. B is called a Vandermonde matrix. It has a very nice structure with 1's, and then a, b, c; a squared, b squared, c squared. It can be bigger, and you'll just have cubes and more letters down here. C is given by the product of these two, an...

ANA RITA PIRES: Hi. Welcome back to recitation. In lecture, you've been learning about the properties of determinants. To remember, there were three main properties, and then seven more that fall out of those three. I'll tell you what these three were. The first one was the determinant of the identity matrix is always equal to 1. If you switch two rows in a matrix, the determinant switches sign. And the determinant is a function of each-- it's a linear function of each row separately. And there's seven more. We'll use them here. Today's problem is about finding the determinants of matrices by using these properties. So here you have four matrices. A has lots of 100's, 200's, and 300's numbers. B is called a Vandermonde matrix. It has a very nice structure with 1's, and then a, b, c; a squared, b squared, c squared. It can be bigger, and you'll just have cubes and more letters down here. C is given by the product of these two, an...

ANA RITA PIRES: Hi there. Welcome to recitation. In lecture, you've been learning about vector spaces whose vectors are actually matrices or functions, and this is what our problem today is about. We have a set of 2 by 3 matrices whose null space contains the vector [2, 1, 1]. And I want you to show that this set is actually a vector subspace of the space of all 2 by 3 matrices. And then, I want you to find a basis for it. When you're done, here is an additional question. What about the set of those 2 by 3 matrices whose column space contains the vector [2, 1]? All right. Hit pause and work on it yourself, and when you're ready, I'll come back and show you how I did it. Hi. I hope you managed to solve it. Let's do it. So, how do we show that something is a vector subspace? Well, there are only two things that we need to check. One is that if two vectors, in this case two matrices, are in that space, then their sum is in that space. And if you take a ve...

ANA RITA PIRES: Hi there. Welcome to recitation. In lecture, you've been learning about vector spaces whose vectors are actually matrices or functions, and this is what our problem today is about. We have a set of 2 by 3 matrices whose null space contains the vector [2, 1, 1]. And I want you to show that this set is actually a vector subspace of the space of all 2 by 3 matrices. And then, I want you to find a basis for it. When you're done, here is an additional question. What about the set of those 2 by 3 matrices whose column space contains the vector [2, 1]? All right. Hit pause and work on it yourself, and when you're ready, I'll come back and show you how I did it. Hi. I hope you managed to solve it. Let's do it. So, how do we show that something is a vector subspace? Well, there are only two things that we need to check. One is that if two vectors, in this case two matrices, are in that space, then their sum is in that space. And if you take a ve...

ANA RITA PIRES: In lecture, you've learned about Gram-Schmidt orthogonalization, and that's what today's problem is about. We have a matrix A, and its columns are a, b, and c. And I want you to find orthonormal vectors q_1, q_2, and q_3 from those three columns. Then I want you to write A as a-- it's QR decomposition where Q is an orthogonal matrix, and R is an upper triangular matrix. Remember, an orthogonal matrix is a matrix whose columns are orthonormal vectors. Work on it for a little while, hit pause, and when you're ready I'll come back and we'll do it together. Did you manage to solve that all right? Well let's start solving it together. So Gram-Schmidt orthogonalization, as you should remember from lecture, consists of the following. At each step, you find your orthonormal vector by taking the vector that you started with, a, b, or c in this case, and making it orthonormal to the previous ones. Let's actually do it. We want to ...

ANA RITA PIRES: In lecture, you've learned about Gram-Schmidt orthogonalization, and that's what today's problem is about. We have a matrix A, and its columns are a, b, and c. And I want you to find orthonormal vectors q_1, q_2, and q_3 from those three columns. Then I want you to write A as a-- it's QR decomposition where Q is an orthogonal matrix, and R is an upper triangular matrix. Remember, an orthogonal matrix is a matrix whose columns are orthonormal vectors. Work on it for a little while, hit pause, and when you're ready I'll come back and we'll do it together. Did you manage to solve that all right? Well let's start solving it together. So Gram-Schmidt orthogonalization, as you should remember from lecture, consists of the following. At each step, you find your orthonormal vector by taking the vector that you started with, a, b, or c in this case, and making it orthonormal to the previous ones. Let's actually do it. We want to ...