True/False: If A is an invertible m x m matrix and B is an m x...

True/False: If A is an invertible m x m matrix and B is an m x n matrix, then AB and B have the same null space.

(Hint: AB and B have the same null space means that ABX = 0 if and only if BX = 0, where X is in Rⁿ.)

Solutions

Expert Solution

We are given that A is an invertible m x m matrix and B is an m x n matrix. We have to verify the veracity of the following statement:

If A is an invertible m x m matrix and B is an m x n matrix, then AB and B have the same null space.

We claim that this statement is true and we shall provide a proof for the above statement.

Proof : We are provided with the following hint.

AB and B have the same null space means that ABX = 0 if and only if BX = 0, where X is in

First we consider the case BX = 0 for . So if X is in the null space of B, we have

Now we consider the matrix-vector product ABX .

Thus we have if BX = 0 then obviously ABX = 0.

Now consider the other way around. Suppose we are given that ABX = 0 for some X in , that is X is in the null space of AB.

Since A is an invertible matrix we can multiply both sides of the equation by its inverse.

where I is the m x m identity matrix.

Finally we get

So we have shown that if ABX =0 then BX = 0.

Thus we have successfully proven that ABX = 0 if and only if BX = 0 which in turn means that the null space of AB and B are one and the same.

Colby Messinger answered 4 months ago

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