Relaue solving systems of the form AX B to one of your previ

Relaue solving systems of the form AX B to one of your previous Engineering, Stat or (Math for education )courses. Show a specific detailed application. (no computations are needed here)

Solution

Invertible matrix:

If A is an invertible matrix then pre-multiplying both the sides by A^-1, we get the solution x = A^-1B.

If A is not invertible, then Ax = B will have either no solution, or an infinite number of solutions.

If B = 0 then the set of all solution to Ax = 0 is the nullspace of A.

Let A = [A₁|A₂| · · · |A_n] be an m × n matrix, with columns A₁, A₂, . . . , A_n. Then a solution x = (x₁, x₂,…,x_n )^T to the matrix equation Ax = B satisfies x₁ A₁ + x₂ A₂ + · · · + x_n A_n = Bb. The equation Ax = b has a solution exactly when b is a linear combination of the columns of A. Thus, For any matrix A and right hand side vector B, the equation Ax = B has a solution implies that x is in the column space R(A) of A.

Suppose that there are two vectors y and z which are solutions of the equation Ax = B, i.e. Ay = B and Az = B. Then, on subtracting the first equation from the second equation we get A(z y) = Az Ay = B B = 0. Hence, if y and z both solve Ax = B, then z y N (A) (the nullspace of A). In other words, if y and z both solve Ax = B, then z y = n for some vector n N (A). Moving y to the right hand side, we get the rule “If y and z are any two solutions to Ax = B, then z = y + n for some vector n N (A).

General Solution:

Let x_p be a particular solution to Ax = B. Then the general solution to Ax = B is given by x_p + x_n where x_n is a vector in the nullspace of A. (In other words, each vector of the form x = x_p + x_n solves Ax = B, and every vector x which solves Ax = B can be written in this form.)

Solvability Criterion:

On performing Gaussian elimination to transform [A|b] to [U|c], the system of equations Ax = B has a solution if and only if for every zero row of U the corresponding entry of c is zero.

Column rank? Row Rank:

The column rank of a matrix A is the dimension of the subspace spanned by the columns of A. Similarly, the row rank is the dimension of the subspace of the space spanned by the rows of A. The row rank and the column rank of a matrix A are equal.

All of the following are equivalent criteria for a matrix A to have full column rank:

1. All columns of A are pivot columns.

2. There are no free variables or special solutions.

3. The nullspace N(A) contains only the zero vector x = 0.

4. If Ax = B has a solution (it may not), then it has only one solution.

All of the following properties for a m x n matrix A mean the matrix has full row rank:

1. All rows have pivots, and R, the RREF of A, has no zero rows.

2. Ax = B has a solution for every right side B.

3. The column space is the whole space R ^m

4. There are n — r = n — m special solution in the nullspace of A.

If a square matrix A is invertible, it has both full column rank and full row rank.