The rank of coefficient matrix A of three variable non-homogeneous equations AX = B is 2, and the three solution vectors a, B, C satisfy a + B = (3.1. - 1) a + C = (2.0. - 2), and the general solution is obtained

The rank of coefficient matrix A of three variable non-homogeneous equations AX = B is 2, and the three solution vectors a, B, C satisfy a + B = (3.1. - 1) a + C = (2.0. - 2), and the general solution is obtained

The rank of coefficient matrix A is 2, so the fundamental solution system of homogeneous equation has 3-2 = 1 vector
(a + b) - (a + C) = B-C, which is the solution of the second equation
So we find the basic solution system
（3,1,-1）-（2,0,-2）=（1,1,1）
Since (a + B + A + C) / 4 is the solution of the non-homogeneous equation, we find a special solution
[（3,1,-1）+（2,0,-2）]/4=（5/4,1/4,-3/4）
To sum up, the general explanation is as follows:
k(1,1,1) + （5/4,1/4,-3/4）

If AX = ay and a is not equal to 0, then x = y. please give an example
Note that a ≠ 0

Here's a simple one
A =
1 0
0 0
X =
one
one
Y =
one
2 (the key is here, you can let any number go)
Then AX = ay=
one
0
Obviously x ≠ y

Whether there is a solution to the system of quaternion linear equations is discussed by using the augmented matrix

The steps to determine whether a system of linear equations with four variables has a solution by using the augmented matrix are as follows: first, find out its coefficient matrix and augmented matrix (the augmented matrix is to add a column on the right side of the coefficient matrix, which is the value on the right side of the equal sign of the system of linear equations), and then find out their respective ranks

If the system of equations has infinitely many solutions, what conditions does the augmented matrix satisfy

r(A,b)=r(A)

In linear algebra, what is the difference between the rank of augmented matrix and that of coefficient matrix?

The rank of the augmented matrix represents the number of solution vectors of the corresponding non-homogeneous equation! The rank of the coefficient matrix represents the number of solution vectors of the corresponding homogeneous equation!

What is the meaning of the rank of the augmented matrix? For example, what is the specific meaning of the rank of the augmented matrix in the equations of three planes

The necessary and sufficient condition for a system of linear equations to have a solution is that its coefficient matrix has the same rank as its augmented matrix. It should be pointed out that this discriminant adjustment is consistent with the elimination method. We know that the first step in solving the system of linear equations by the elimination method is to use the elementary row transformation to transform the augmented matrix into a ladder type. This ladder type matrix may have two types after properly adjusting the order of the first n columns Situation: one is the last behavior which is not all zero 0 d(r+1)
Or 0 c（rr)…… c（rn） d（r）
In fact, if the last column of the ladder matrix is removed, it is the ladder form of the coefficient matrix of the system of linear equations through elementary transformation. That is to say, if the rank of the coefficient matrix and the augmented matrix is the same, the system of equations has a solution; if the rank of the augmented matrix is equal to the rank of the coefficient matrix plus 1, the system of equations has no solution
Let's look at the equations a11x1 + a12x2 + a13x3 = B1; a21x1 + a22x2 + a23x3 = B2;
His coefficient matrix is a = {a11 A12 A13 augmented matrix B {a11 A12 A13 B1
a21 a22 a23} a21 a22 a23 b2}
Their rank may be 1 or 2
R (a) = 1, R (b) = 1. This means that the two lines of a are proportional, so the two planes are parallel. Because the two lines of B are also proportional, the two planes coincide, and the equations have solutions
2. R (a) = 1, R (b) = 2. It means that two planes are parallel but not coincident
3. R (a) = 2, then R (b) must be 2. Geometrically, the two planes are not parallel, so they must intersect. There is no solution to the equations
Copy books, as typing practice

Matrix equivalence and vector group equivalence
A. B is a square matrix of order n, P and Q are invertible matrices of order n. if B = PAQ, then the row (column) vector group of a and the row (column) vector group of B are equivalent. Why not?

Invertible matrix does not change the rank of matrix, that is, R (b) = R (PAQ) = R (a), so row (column) rank of a = row (column) rank of B
However, the row (column) vector groups of a and B may not be linear to each other
Remember the following two related knowledge points:
1. If B = PA, then the row vector groups of a and B are equivalent
If B = AQ, then the set of column vectors of a and B is equivalent
But if B = PAQ, there is no corresponding conclusion
2. If B = PA, then the column vector group of B has the same linear relationship with the corresponding column vector group of A
The primary row transformation does not change the linear relationship of the column vector group

Mr. Liu, what's the difference between the equivalence of matrix and the equivalence of vector group?
The equivalence of a matrix must be equal rank, and equal rank must be equal, so it doesn't need to be of the same type? Why does the book directly say that the necessary and sufficient condition for the equivalence of a matrix is equal rank?

The premise of matrix equivalence is the same type
If and only if the rank is the same
When reading a book, you should pay attention to the context, which is considered under the condition of the same type
The necessary and sufficient condition for equivalence of vector group is R (a) = R (a, b) = R (b)

Linear Algebra: isn't vector group equivalence the same as matrix equivalence
Similarly, R (a) = R (b), is it not equal to R (a, b)

If two n-dimensional vector groups are equivalent, then the rank of matrices A and B composed of them as column vectors is equal, but not necessarily equivalent, because the number of columns of these two matrices may be different. For example, a matrix with five rows and three columns and a matrix with five rows and four columns are not equivalent at all

On the question of linear algebra, why can't the equivalence of vector group be equal to that of corresponding matrix
On the question of linear algebra, why can't the equivalence of vector groups be equal to the equivalence of corresponding matrices, but should be defined as two vector groups can be expressed linearly with each other. What's the significance of this? In terms of use?

I might go deeper:
1: Vector group equivalence and matrix equivalence can not be mutually deduced without other special instructions. A): vector group equivalence cannot deduce matrix equivalence because the vector group equivalence of two matrices can not guarantee that the two matrices are of the same type, But most of the time, the corresponding matrices of these two vector groups are different. B): matrix equivalence can not deduce the equivalence of vector group because: the equivalence of matrix may use row transformation and column transformation at the same time, while the equivalence of vector group only allows the single use of row transformation (called row vector group equivalence) or single column transformation (called column vector group equivalence), The equivalence of two moment arrays must be the equivalence of column vector groups (note that the rows or columns before matrix equivalence cannot be omitted)
2: For the function of vector equivalence: a) from the point of view of solving equations, vector equivalence means that the two equations have the same solution, but simple matrix equivalence can not guarantee this. B) another meaning of introducing vector equivalence is to consider that the matrix can only express finite order (because the matrix must write the elements one by one), Although its number is also infinite, the function of this infinite array can be perfectly expressed by a finite, that is, its maximum independent system. Thus, the problem of infinity is solved. For example, if a system of equations has infinite solutions, it can not be expressed by matrix, In fact, any vector group represents the set of multiple vectors in a certain space. The simplest example is the spatial coordinate system of senior high school. When I was in senior high school, I met five vectors in a topic, such as vector a, B, C, D, and so on, e. Even more. But think about these five vectors can be represented by only three coordinate unit vectors I, J, K. These three coordinate axes are also called bases (I learned in high school), and I said in high school that as long as there are three vectors that are not coplanar, they can become a group of bases in three-dimensional space. Therefore, no matter how many vectors are, the bases of their space are limited, I don't care how many vectors you have. Anyway, I can express them with bases. A group of bases is actually the largest independent group. The number of bases (i.e. the number of coordinate axes required) is the rank of the vector group (so you can understand: Oh! The original rank represents the number of coordinate axes required to express all vectors in the vector group), We can learn many hard to remember theorems in the generation without line of sight
All hands, or cell phone. Tired