Question

Solvability of linear equations. Let
y EW
be arbitrary. Wish to find such that and is linear. Find conditions on T such that there is a solution to   for each
y EW
and the solution is unique.

Answer 1

Solvability of linear equation .

We need to find an condition on such that for all
YEW
there exist $x\in V$ such that
T(I) = y
.

Suppose

Dim(V) = m . Dim(W) =n

Suppose
{41:42: ..... Yn}
be a basis of so they are linearly independent . If
21.12....,
are preimage of
91. 92. .... Yn
repectively then  
21.12....,
is also an linearly independent set in .

As contains a set of n vectors which are linearly independent so  
Dim(V) >n
.

→ DimV) > Dim(W

As  
{41:42: ..... Yn}
be a basis of so any element of can be written as linear combination of elements of  
{41:42: ..... Yn}
so  
Tr,), Try),....Τα.)
are innearly independent set in W .

Rank T=n

Rank T = Dim W

Hence the required condition on for which
T(I) = y
has a solution for all
YEW
is ,

Rank T = Dim W
and
Dim(V) > Dim(W)
.

.

.

.

If you any doubt please comment .