A system of 700 linear equations in 500 unknowns is to be solved. The rank of th

Question

A system of 700 linear equations in 500 unknowns is to be solved. The rank of the coefficient matrix is known to be equal to 490. how many free variables will the associated homogeneous system have? will there be any non-zero homogeneous solutions? will the equation system have a solution for any right hand side constants? what is the maximum number of linearly independent equations? what is the maximum number of linearly independent columns of the coefficient matrix? what is the dimension of the null space of the coefficient matrix?

Explanation / Answer

The coefficient of matrix A is of the order 700 x 500

a) Rank(A) = r = number of non zero rows in its echeleon form = 490

Rank (A|0) = number of non zero rows in its echeleon form = 490

Rank (A|0) = Rank(A) < Number of unknowns = n = 500

So, this has infinite number of solutions

Let's assign random values to (n-r) = 500 - 490 = 10 so that the system is consistent

Thus, 10 free variables.

b) Yes, there will be non zero homogeneous solutions.

c) No, because the system has a solution only if Rank (A|B) = Rank (A)

d) The maximum number of linearly independent equations are Rank (A) = 490

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