A is an m×n matrix. Check the true statements below:A. The kernel of a linear transformation is a vector space. B. If the equation Ax=b is consistent, then Col(A) is Rm. C. The null space of an m×n matrix is in Rm. D. The column space of A is the range of the mapping x→Ax. E. Col(A) is the set of all vectors that can be written as Ax for some x. F. The null space of A is the solution set of the equation Ax=0.

Results are (1) True. (2) False. (3) False. (4) True. (5) True. (6) True.

Step-by-step explanation:

Given A is an matrix.  Let  be the corresponding linear transformationover the field F and be identity vector in V. Now if .

(1) The kernel of a linear transformation is a vector space : True.

Let , then,

hence the kernel is closed under addition.

Let , then

and thus Ket(T) is closed under multiplication

Finally, fore all vectors ,

Thus Ker(T) is a subspace.

(2) If the equation Ax=b is consistent, then Col(A) is : False

if the equation Ax=b is consistent, then Col(A) must be consistent for all b.

(3) The null space of an mxn matrix is in

: False

The null space that is dimension of solution space of an m x n matrix is always in .

(4) The column space of A is the range of the mapping

: True.

(5) Col(A) is the set of all vectors that can be written as Ax for some x. : True.

Here Ax will give a linear combination of column of A as a weights of x.

(6) The null space of A is the solution set of the equation Ax=0.

: True