Question

Let A be an mxn matrix. Use the steps below to show that a vector x in Rn satisfies Ax = 0 if and only if A ᵀAx=0. This will show that Nul A = Nul AᵀA.

Show that if Ax = 0, then AᵀAx=0.

Answer 1

If Ax=0, then AᵀAx=0 because multiplying both sides of Ax=0 by Aᵀ results in Aᵀ(Ax) or Aᵀ*0, which is also the null vector, showing that Nul A equals Nul AᵀA.

Step-by-step explanation:

The question involves understanding the relationship between the null space (Nul) of a matrix A and the matrix product AₚA. To demonstrate that Ax = 0 if and only if AₚAx=0, we must show that if Ax equals the null vector, then AₚAx will also equal the null vector.

Let's consider a matrix A which is an mxn matrix and a vector x in Rⁿ. If Ax = 0, it means that x is in the null space of A, i.e., Nul A. Now, multiplying both sides of the equation Ax = 0 by Aₚ, the transpose of A, we obtain AₚAx = Aₚ0.

Since the product of any matrix with the null vector is also a null vector, Aₚ0 = 0 and therefore, we have AₚAx = 0. This demonstrates that if Ax = 0, then it is also true that AₚAx = 0. Consequently, any vector x that is in the null space of A (Nul A) will also be in the null space of AₚA (Nul AₚA), showing that Nul A = Nul AₚA.