Question

Let T be a linear transformation from vector space V onto Wand dim V = dim W, then
A. Rank (7)<dim V
B. Rank (T) = dim V
C. Rank (7) > dim V
D. Rank (T) = Nullity (T)
urgent

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Answer 1

B.    Rank (T) = dim V .

Given:

T be a linear transformation from vector space V onto W

 and  dim V = dim W.

To Choose:

A. Rank (7)<dim V

B. Rank (T) = dim V

C. Rank (7) > dim V

D. Rank (T) = Nullity (T)

Solution:

Here, it is given that - dim V = dim W

From the dimension theorem;

dim(V)=nullity(T)+rank(T)

Here,  T be a linear transformation from vector space V onto W and

dim V = dim W therefore, T is one-one .

so, Nullity(T)=0

Now, dim(V)=nullity(T) + rank(T)

        dim(V)= 0 + rank(T)

        dim(V)= rank(T)

Hence, Rank(T) = Dim(V).