Learning and organizing linear algebra 6: Vectors and matrices in detail, what is a matrix? (Draft ----- not completed)

 

 43 Matrix

4.1 Matrix

 4 Organize the Internet and summarize some views on the essence of linear algebra

The essence of the matrix is ​​rotation and scaling

• The number 0 in the matrix
• The number 1 in the matrix means no scaling
• The number 2 in the matrix, etc., means scaling
• The number -3 in the matrix means zoom -3 times, and reverse
• the position of the number in the matrix
• matrix split into column vectors

For example, in the following matrix, if the unit matrix is ​​placed on the left, it means that the elements in the first row of the matrix are *1, and the elements in the second row are *1, which means that nothing is actually done.

 1  0 

 0  1 

A matrix is ​​a simplified representation of a column vector

• A matrix is ​​a simplified form of writing multiple column vectors together
• That is to say

1. The following are equivalent:
2. Matrix addition is equivalent to adding multiple column vectors separately
3. Matrix multiplication is equivalent to multiplying multiple column vectors separately

4.2 Dimensions of matrices

• (a1,a2) is 2-dimensional
• (a1,a2,a3) is 3 dimensional
• (a1,a2,a3... an) is n-dimensional

column vector of matrix

• Each column vector of the matrix
• Both represent that the base ei in this direction has reached the position of the corresponding column vector.
• for example

Flat concept of matrix

That is, the meaning of linear growth when the matrix needs to be

For example, matrix 10, 10 matrices cannot be reduced to 90, but must be 100

 3 Elementary transformations of matrices (leading out the concept of matrix equivalence)

• If two matrices can be equal after a finite number of elementary changes, then the two matrices are equivalent
• The elementary row transformation and elementary column transformation of a matrix are collectively called the elementary transformation of a matrix.
• Elementary Row Transformation of Matrix

1. Switch the two rows of the matrix
2. Multiply all elements of a row of the matrix by a non-zero number k
3. Multiply all the elements of a certain row of the matrix by a number k and add them to the corresponding elements of another row

• Elementary Column Transformation of Matrix

1. Swap the two columns of the matrix
2. Multiply all elements of a column of the matrix by a non-zero number k
3. Multiply all the elements of a certain column of the matrix by a number k and add them to the corresponding elements of another column

map of matrix multiplication

Rank of the matrix

Matrix multiplication is not commutative

• A*B != B*A
• A left multiplied by *B != A right multiplied by *B
• Suppose A!=0, B!=0, but there may be A*B=0
• Suppose A!=0, but there may be A*A=0
• If it is known that A*B=C, then B= A-*C , but B != C*A-

Linear algebra, matrix, belongs to algebra, not to geometry,

It is a bit difficult to understand the geometric meaning of matrix multiplication

matrix modulus

3.2 Square matrix

Square matrices have many special properties

reversible

The determinant is also a special calculation method of the equation

3.3 Base

4.2 Basis/basis of matrix

• (a1, a2) is 2-dimensional, corresponding to 2 bases e1, e2
• (a1, a2, a3) is 3-dimensional, corresponding to 3 bases e1, e2
• (a1, a2, a3... ... an) is n-dimensional, corresponding to n bases e1, e2.....en
• For example, a vector (3,2,5) can be considered as the length/scalability of the three bases

1. Base 1, length/stretch on (1,0,0) is 3,
2. The second basis, the length/stretch on (0,1,0) is 2,
3. Base 3, length/stretch on (0,0,1) is 5,

4.2 The base/basis of the matrix can be changed

3.4 Transformation of vectors, two methods

The base remains the same, the coordinates will change (and the shape may also change)

Kibenha/Substituted, the coordinates are unchanged (and the shape cannot be changed at the same time)