Rank of T3
To determine the rank of T3, we need to consider the given information about T1 and T2 and analyze their relationship with T3.

Rank of T1
The rank of T1 is given as 3. The rank of a linear transformation is the dimension of its range. Since the rank of T1 is 3, it means that the range of T1 has a dimension of 3. In other words, T1 maps a 5-dimensional vector space (R5) to a 3-dimensional vector space (R3).

Nullity of T2
The nullity of T2 is given as 3. The nullity of a linear transformation is the dimension of its null space. Since the nullity of T2 is 3, it means that the null space of T2 has a dimension of 3. In other words, T2 maps three linearly independent vectors to the zero vector.

T3.T1 = T2
According to the given information, T3 composed with T1 is equal to T2. In other words, the composition of T3 and T1 results in the same linear transformation as T2. This relationship allows us to analyze the rank of T3.

Rank of T3
Since T3.T1 = T2, we can conclude that the range of T3.T1 is equal to the range of T2. The range of T3.T1 is a subspace of R3, and the range of T2 is also a subspace of R3. Therefore, the range of T3.T1 is contained within the range of T2.

Since the range of T3.T1 is contained within the range of T2, the dimension of the range of T3.T1 is less than or equal to the dimension of the range of T2. Since the rank of T2 is 3 (as given by the nullity of T2), it means that the range of T2 has a dimension of 3.

Therefore, the rank of T3 is at most 3. It is possible for the rank of T3 to be less than 3 if the range of T3.T1 is a proper subspace of the range of T2.

In conclusion, the rank of T3 is at most 3, and it can be less than 3 depending on the specific properties of T3.