Respuesta :
Answer:
Results are (1) True. (2) False. (3) False. (4) True. (5) True. (6) True.
Step-by-step explanation:
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
