a) To graph: the ker(A),ker(A)^bot and im(A^T) ​ b) To find: the relationship between im(A^T) and ker(A).

Brittney Lord 2021-02-27 Answered

a) To graph: the ker(A),(kerA)andim(AT)
b) To find: the relationship between im (AT) and ker (A).
c) To find: the relationship between ker(A) and solution set S
d) To find x0 at the intersection of ker(A)and(kerA)
e) To find: the lengths of x0 compared to the other vectors in S

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Expert Answer

Corben Pittman
Answered 2021-02-28 Author has 83 answers

a) Given:
The image of linear transformation T(x)=A(x) is the span of the column vectors of A.
The kernel of a linear transformation T(x)=A(x)omRmRn consists of all zeros of the transformation.
That is the solution of the equation T(x)=A(x)=0.
We denote the kernel of T by ker (A) or ker (T) and it is the solution set of the linear system A(x)=0.
Graph:
image
Interpretation:
Consider the linear system A(x)=bforA=[1326]andb=[1020].
A(x)=0
[1326][x1x2]=0
x1=3t,x2=t
t=parametr
[x1x2]=[3tt]=[31]t
Thus, ker (A) is the line spanned by [31]R2.
Consider im (A) which is spanned by u1=[12]. Then,
(kerA)=ker(A)proj(kerA)
(kerA)=ker(A)(u1.kerA)u1
(kerA)=[31][[12][31]][12]
(kerA)=[31]+[12]
(kerA)=[23]
Consider AT=[1236].
Since [1236]=

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