Rank of T:

To determine the rank of the linear transformation T, we need to find the dimension of the range of T. The range of T is the set of all possible outputs of T for all possible inputs.

Range of T:
The range of T is the set of all possible outputs of T for all possible inputs. In this case, the linear transformation T maps (x, y, z) to (x^2, y^2, 0).

Let's consider the possible outputs of T:
- The first coordinate of the output is x^2, which can take any real value.
- The second coordinate of the output is y^2, which can also take any real value.
- The third coordinate of the output is always 0.

Dimension of the Range:
The dimension of the range is the number of linearly independent vectors that span the range. In this case, since the third coordinate of the output is always 0, it means that the range of T lies entirely in the xy-plane.

Therefore, the range of T is a two-dimensional subspace of R³, spanned by the vectors (1, 0, 0) and (0, 1, 0). These two vectors are linearly independent, as they cannot be expressed as scalar multiples of each other.

Hence, the dimension of the range of T is 2.

Rank of T:
The rank of a linear transformation is defined as the dimension of the range of the transformation. In this case, since the dimension of the range of T is 2, the rank of T is also 2.

Conclusion:
The rank of the linear transformation T, given by T(x, y, z) = (x^2, y^2, 0), is 2. This means that the range of T is a two-dimensional subspace of R³, spanned by the vectors (1, 0, 0) and (0, 1, 0).