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LINAN CHEN: Hi. Welcome back to recitation. In the lecture, you've learned eigenvalues and eigenvectors of a matrix. One of the many important applications of them is solving a higher-order linear differential equation with constant coefficients. A typical example is like what I've written on the board here. y is a function of t, and y and its derivatives satisfy this equation. As you can see, it involves y, y prime, and all the way to its third derivative. So our first goal is to solve this differential equation for its general solution using the method of matrix. So the very first thing that we should do is to find out which matrix we should be working with. So after that, we also want to say something about the explanation of this matrix A*t. We want to find out the first column of this matrix exponential. Why don't you hit the pause now, and try to write down this matrix A by yourself. But before you continue, make sure you come back to this video and chec...

LINAN CHEN: Hi. Welcome back to recitation. In the lecture, you've learned eigenvalues and eigenvectors of a matrix. One of the many important applications of them is solving a higher-order linear differential equation with constant coefficients. A typical example is like what I've written on the board here. y is a function of t, and y and its derivatives satisfy this equation. As you can see, it involves y, y prime, and all the way to its third derivative. So our first goal is to solve this differential equation for its general solution using the method of matrix. So the very first thing that we should do is to find out which matrix we should be working with. So after that, we also want to say something about the explanation of this matrix A*t. We want to find out the first column of this matrix exponential. Why don't you hit the pause now, and try to write down this matrix A by yourself. But before you continue, make sure you come back to this video and chec...

LINAN CHEN: Hi everyone. I'm Linan. Welcome back to recitation. In recent lectures, we have studied the properties of the determinant. And we also derived the formula to compute it. Today we're going to put what we learned into practice by considering these two examples. So we want to find out the determinants of these two 5 by 5 matrices. And as you can see, matrix A has x along this diagonal, and in the first four rows, y to the right of x, except for the last row. And zero entries everywhere else. And matrix B also has x along this diagonal and y everywhere else. Before starting, let me help you review what you can do to compute determinants. Of course, you can carry out elimination to transform your original matrix into upper triangular matrix. Or you can use this big summation formula. Another choice would be you can do it by cofactors. Namely, you can expand your original matrix along any row or any column, and then the determinant is simply given by the dot...

LINAN CHEN: Hello. Welcome back to recitation. I'm sure you are becoming more and more familiar with the determinants of matrices. In the lecture, we also learned the geometric interpretation of the determinant. The absolute value of the determinant of a matrix is simply equal to the volume of the parallelepiped spanned by the row vectors of that matrix. So today, we're going to apply this fact to solve the following problem. I have a tetrahedron, T, in this 3D space. And the vertices of T are given by O, which is the origin, A_1, A_2, and A_3. So I have highlighted this tetrahedron using the blue chalk. So this is T. And our first goal is to compute the volume of T using the determinant. And the second part is: if I fix A_1 and A_2, but move A_3 to another point, A_3 prime, which is given by this coordinate, I ask you to compute the volume again. OK. So since we want to use the fact that the determinant is related to the volume, we have to figure out which volume...

LINAN CHEN: Hello. Welcome back to recitation. I'm sure you are becoming more and more familiar with the determinants of matrices. In the lecture, we also learned the geometric interpretation of the determinant. The absolute value of the determinant of a matrix is simply equal to the volume of the parallelepiped spanned by the row vectors of that matrix. So today, we're going to apply this fact to solve the following problem. I have a tetrahedron, T, in this 3D space. And the vertices of T are given by O, which is the origin, A_1, A_2, and A_3. So I have highlighted this tetrahedron using the blue chalk. So this is T. And our first goal is to compute the volume of T using the determinant. And the second part is: if I fix A_1 and A_2, but move A_3 to another point, A_3 prime, which is given by this coordinate, I ask you to compute the volume again. OK. So since we want to use the fact that the determinant is related to the volume, we have to figure out which volume...

LINAN CHEN: Hi everyone. I'm Linan. Welcome back to recitation. In recent lectures, we have studied the properties of the determinant. And we also derived the formula to compute it. Today we're going to put what we learned into practice by considering these two examples. So we want to find out the determinants of these two 5 by 5 matrices. And as you can see, matrix A has x along this diagonal, and in the first four rows, y to the right of x, except for the last row. And zero entries everywhere else. And matrix B also has x along this diagonal and y everywhere else. Before starting, let me help you review what you can do to compute determinants. Of course, you can carry out elimination to transform your original matrix into upper triangular matrix. Or you can use this big summation formula. Another choice would be you can do it by cofactors. Namely, you can expand your original matrix along any row or any column, and then the determinant is simply given by the dot...