)tiny-tim said:hi junsugal!
(do you mean Ker(A)? it's spelled "kernel", not "colonel" … a kernel is the soft part inside the shell of a nut!
it means you have to prove that there exists an x such that x is in Nul(A) but not Col(A)
(i think the question should have stipulated that A is non-zero)
junsugal said:
junsugal said:
Being a proper subset means that Col(A) is a subset of Nul(A), but not equal to it.
Col(A) and Nul(A) are both subspaces of the vector space of A. They are complementary subspaces, meaning that their intersection is only the zero vector.
To prove this, we need to show that every vector in Col(A) is also in Nul(A), but there exists at least one vector in Nul(A) that is not in Col(A).
This relationship tells us that the column space and null space of a matrix are not equal, and that there exists a non-trivial solution to the equation Ax=0.
Yes, consider the matrix A = [1 0 0; 0 1 0; 0 0 0]. The column space of this matrix is the xy-plane, while the null space is the z-axis. Therefore, Col(A) is a proper subset of Nul(A).