The relation between the form of solution of AX = B and the rank of matrix A of non homogeneous linear equations? What I need to know is the form of the solution of the system of equations, such as k1x1 + k2x2 +... + knxn + y, x1, X2... Xn is the basic solution system, I can't determine the specific value of n~

The relation between the form of solution of AX = B and the rank of matrix A of non homogeneous linear equations? What I need to know is the form of the solution of the system of equations, such as k1x1 + k2x2 +... + knxn + y, x1, X2... Xn is the basic solution system, I can't determine the specific value of n~

The number of solution vectors contained in the basic solution system is: the number of unknowns - the rank of A
Number of unknowns = number of columns of a

Let a be an mxn matrix, B be an nxm matrix, and satisfy AB = e, then the following is the option
To solve the problem of the process steps, had better be more detailed, thank you
(A) The column vector group of a is linearly independent, and the row vector group of B is linearly independent
(B) The group of column vectors of a is linearly independent, and the group of column vectors of B is linearly independent
(C) The row vector group of a is linearly independent, and the column vector group of B is linearly independent
(D) The row vectors of a are linearly independent, and the row vectors of B are linearly independent

From ab = e, R (AB) = R (E) = m
So m = R (AB)

Let a be mxn matrix, B be nxm matrix, and M > N, and prove det (AB) = 0

Because R (AB)

Let η 1 and η 2 be two different solutions of the non-homogeneous linear system AX = B (a is m × n matrix), and ξ be the non-zero solution of the corresponding homogeneous linear system AX = 0. It is proved that: (1) the vector systems η 1, η 1 - η 2 are linearly independent; (2) if the rank r (a) = n-1, then the vector systems ξ, η 1, η 2 are linearly related

The following: (1) let K1, η 1 + K2 (η 1-η 1 + K2 (η 1-η 2) = 0, then K 1a, η 1 + K 2A (η 1-η 2) = 0, then K 1a, η 1 + K 2A (η 1, η 2) is known to be two different solutions of the nonhomogeneous linear equations AX = B, which are two different solutions of AX = B, and known that η 1 and η2 are two different solutions of the nonhomogeneous linear equations AX = B, ax = b = b = 0, so a η 1 = a η 1 = a η 1 = a η 2 = a = a η 2 = a η 2 = a η 2 = a η 2 = a η 2 = a η 2 = a η 2 = a η 2 = a = a = a = a = a = a = R (a) r (a) N-R (a) N-R (a) = N-R) n-n-n-n-n-the basic solution system is only There is a solution vector, which is a basic solution system of AX = 0, and η 1 - η 2 is a non-zero solution of AX = 0. There is a linear correlation between ξ and η 1 - η 2, i.e. there is a number k such that η 1 - η 2 = k ξ ξ K ξ + η 1 - η 2 = 0, i.e. there is a linear correlation among vector groups ξ, η 1 and η 2

Finding the general solution of linear equations AX = b
Let a be a third-order square matrix, R (a) = 2, ax = B have three solutions x1, X2, X3. X1 = [1,2,3] ^ t, X2 + X3 = (2,3,4) ^ t, then what is the general solution of the linear equation system AX = B?
How to analyze. No idea at all. Ask for explanation

Because R (a) = 2
So the fundamental solution system of AX = 0 contains 3-R (a) = 1 solution vector
So 2x1 - (x2 + x3) = 2 (1,2,3) ^ t - (2,3,4) ^ t = (0,1,2) ^ t is the basic solution system of AX = 0
And X1 = [1,2,3] ^ t is the special solution of AX = B
So the general solution of AX = B is (1,2,3) ^ t + K (0,1,2) ^ t

If a is an M * 4 matrix, R (a) = 3, and the sum of elements in each row of a is 0, then the general solution of the homogeneous linear system AX = 0 is?

If R (a) = 3, there is only one basic solution system of the general solution
So [1,1,1] ^ t is a solution of the equation
So the general solution of the equation is k [1,1,1,1] ^ t

For the n-order square matrix A, the homogeneous linear equations AX = 0 have two linearly independent solution vectors, and a * is the adjoint matrix of A
The solutions of AX = 0 are all solutions of a * x = 0

Let x1, X2 be a with two independent solutions, then s = N-R (a)
r（A）=n-2〈n-1
Then R (a *) = 0, i.e
A*=0
So x1, X2 are also
The solution of a * x = 0

Let α 1, α 2 and α 3 be a basic solution system of homogeneous linear equations AX = 0. It is proved that α 1, α 1 + α 2 and α 2 + α 3 are also the basic solution system of AX = 0

It is proved that: (α 1, α 1 + α 2, α 2 + α 3) = (α 1, α 2, α 3) P
P =
1 1 0
0 1 1
0 0 1
Because | P | = 1 ≠ 0, P is reversible
So α 1, α 2, α 3 are equivalent to α 1, α 1 + α 2, α 2 + α 3
So r (α 1, α 1 + α 2, α 2 + α 3) = R (α 1, α 2, α 3) = 3
The solution of AX = 0 can be expressed linearly by α 1, α 1 + α 2, α 2 + α 3
Therefore, α 1, α 1 + α 2, α 2 + α 3 are the fundamental solutions of AX = 0

Let a be a real square matrix of order n, and let X and B be n-ary sequence vectors over the field of real numbers. It is proved that the sufficient and necessary condition for the linear equation system AX = B to have solutions is that B is orthogonal to the solution space W of the equation system a'x = 0

Let α be any vector in W
Then a 'α = 0
Then α is orthogonal to the row vector of a '
That is, α is orthogonal to the column vector of A
That is to say, W is composed of vectors orthogonal to the column vectors of A
B is orthogonal to W
B is a linear combination of the column vectors of A
Ax = B has solution

Let the adjoint matrix A * ≠ 0 of the n-order matrix A, if ξ 1, ξ 2, ξ 3 and ξ 4 are mutually unequal solutions of the non-homogeneous linear equation system AX = B, then the basic solution system ()
A. There is no B. there is only one nonzero solution vector C. There are two linearly independent solution vectors d. There are three linearly independent solution vectors

∵ A is a matrix of order n, ax = 0 and Ax = B, containing n unknowns. Therefore, the number of vectors in the basic solution system of AX = 0 is N-R (a), and R (a *) = n, R (a) = N1, R (a) = n − 10, 0 ≤ R (a) ≤ n − 2. It is known that a * ≠ 0, then R (a) is equal to N or n-1, and Ax = B has mutually unequal solutions, that is, the solutions are not unique, so: R (a) = n-1, so ax = 0 the solution vector in the basic solution system The number of is: N-R (a) = 1, that is: B