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.

Given A is an [tex]m\times n[/tex] matrix. Let [tex]T :U\to V[/tex] be the corresponding linear transformationover the field F and [tex]\theta[/tex] be identity vector in V. Now if [tex]x\in Ker( T)\implies T(x)=\theta[/tex].

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

Let [tex]x,y\in Ker( T)[/tex], then,

[tex]T(x+y)=T(x)+T(y)=\theta+\theta=\theta\impies x+y\in Ker( T)[/tex]

hence the kernel is closed under addition.

Let [tex]\lambda\in F, x\in Ker( T)[/tex], then

[tex]T(\lambda x)=\lambda T(x)=\lambda\times \theta=\theta[/tex]

[tex]\lambda x\in Ker(T)[/tex] and thus Ket(T) is closed under multiplication

Finally, fore all vectors [tex]u\in U[/tex],

[tex]T(\theta)=T(\theta+(-\theta))=T(\theta)+T(-\theta)=T(\theta)-T(\theta)=\theta[/tex]

[tex]\implies \theta\in Ker(T)[/tex]

Thus Ker(T) is a subspace.

(2) If the equation Ax=b is consistent, then Col(A) is [tex]\mathbb R^m[/tex] : 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 [tex]\mathbb R^m[/tex]

: False

The null space that is dimension of solution space of an m x n matrix is always in [tex]\mathbb R^n[/tex].

(4) The column space of A is the range of the mapping [tex]x\to Ax[/tex]

: 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