Course Address: MIT Open Course Linear Algebra MIT 18.06 Linear Algebra, Spring 2005 Chinese and English subtitles
1. Matrix multiplication
example:
The first one, in Tongji textbooks, is to multiply rows by columns.
C21 = element in row 2 of A, multiplied by element in column 1 of B, accumulating one by one = 3x3 + 4x2 = 17 The
subscript of C corresponds to the row and column of AB, and is calculated one by one.
The second and third use the concept of linear combination in the first lesson. Linear combination is completely different from the multiplication in the textbook, and it took some time to turn the corner.
Review the linear combination multiplication of the following: the
multiplication process in the figure is described as a linear combination of the columns of D, and E indicates how the columns of D are combined, that is to say, F = 1 first column of D + 2 second columns of D Columns, 1 and 2 are provided by E.
When E is expanded to a matrix instead of a column vector, F will also become a matrix. At this time, matrix multiplication can be described as a linear combination of the first column of F is all the columns of matrix D and the first column of E. As for the remaining columns, the combination of D and F and other columns can be obtained.
Example C = AB:
The whole process is described as, the column of C is a linear combination of the columns of A.
The third kind is the same, change to a linear combination of rows, which is how many rows + how many rows:
C = AB:
Note that the row vector is on the left, and the column vector is on the right. Different positions represent different row and column transformations, so the first The three lines described as C are linear combinations of the lines of B.
The fourth type, the sum of the rows of AB = column xB of A:
I understand.
The fifth type, divided into blocks, understand
2. Guass-Jordan elimination
The augmented matrix [A | I] at the end of the video can be abstracted as a block in matrix multiplication. The left block is the matrix A, and the right block is the identity matrix I. By default, A is invertible.
In the process of turning block A into I, all the multiplication and subtraction operations on A can be combined to multiply an elementary matrix E (elimination matrix, taught in Lesson 2), that is to say EA = I, obviously E It is the inverse of A, then EI = E = A^-1.
Therefore, the transformation process of the augmented matrix can be understood as E[A | I] = [EA |EI] = [I | A^-1].
It also explains why the matrix A to be inverted and its identity matrix I are put together [A | I], and when A is converted into I, I will become A^-1.