Rank of a Linear Transformation

The rank of a linear transformation measures the dimension of the vector space spanned by the image (or range) of the transformation. In other words, it represents the number of linearly independent vectors in the image.

Given Information

Let T: R⁷ → R⁷ be a linear transformation such that T² = 0.

Explanation

To determine the rank of T, we need to find the dimension of the image of T. Since T² = 0, it means that the composition of T with itself always results in the zero transformation. This implies that the image of T is contained in the null space (or kernel) of T.

Null Space of T

The null space of T, denoted as null(T), consists of all vectors in the domain of T that are mapped to the zero vector in the codomain. In this case, null(T) is the set of vectors in R⁷ that are mapped to the zero vector.

Rank-Nullity Theorem

According to the rank-nullity theorem, the dimension of the domain of T is equal to the sum of the rank of T and the dimension of null(T). In mathematical notation, this can be expressed as:

dim(R⁷) = rank(T) + dim(null(T))

Since the domain and codomain of T are both R⁷, the dimension of R⁷ is 7. Therefore, we can rewrite the equation as:

7 = rank(T) + dim(null(T))

Rank of T

Since the rank of T is the dimension of the image of T, which is contained in the null space of T, we can conclude that the rank of T is less than or equal to the dimension of null(T).

Conclusion

In this case, since T² = 0, it implies that the image of T is contained entirely within the null space of T. Therefore, the rank of T is 0.