Transformations of rank one

We conclude our discussion of rank by a description of the matrices of linear transformations of rank .

Theorem 1. If a linear transformation on a finite-dimensional vector space is such that (that is, or ), then the elements of the matrix of have the form in every coordinate system; conversely if the matrix of has this form in some one coordinate system, then .

Proof. If , then , and the statement is trivial. If , that is, is one-dimensional, then there exists in a non-zero vector (a basis in ) such that every vector in is a multiple of . Hence, for every , where the scalar coefficient ( ) depends, of course, on . The linearity of implies that is a linear functional on . Let be a basis in , and let be the corresponding matrix of , so that If is the dual basis in , then (cf. Section: Adjoints of projections , (2)) In the present case in other words, we may take and .

Conversely, suppose that in a fixed coordinate system the matrix of is such that . We may find a linear functional such that , and we may define a vector by . The linear transformation defined by is clearly of rank one (unless, of course, for all and ), and its matrix in the coordinate system is given by (where is the dual basis of ). Hence and, since and have the same matrix in one coordinate system, it follows that . This concludes the proof of the theorem. ◻

The following theorem sometimes makes it possible to apply Theorem 1 to obtain results about an arbitrary linear transformation.

Theorem 2. If is a linear transformation of rank on a finite-dimensional vector space , then may be written as the sum of transformations of rank one.

Proof. Since has dimension , we may find vectors that form a basis for . It follows that, for every vector in , we have where each depends, of course, on ; we write . It is easy to see that is a linear functional. In terms of these we define, for each , a linear transformation by . It follows that each has rank one and . (Compare this result with Section: Linear transformations , example (2).) ◻

A slight refinement of the proof just given yields the following result.

Theorem 3. Corresponding to any linear transformation on a finitedimensional vector space there is an invertible linear transformation for which is a projection.

Proof. Let and , respectively, be the range and the null-space of , and let be a basis for . Let be vectors such that is a basis for . Since is in for , we may find vectors such that ; finally, we choose a basis for , which we may denoted by . We assert that is a basis for . We need, of course, to prove only that the ’s are linearly independent. For this purpose we suppose that ; then we have (remembering that for the vector belongs to ) whence . Consequently ; the linear independence of shows that the remaining ’s must also vanish.

A linear transformation , of the kind whose existence we asserted, is now determined by the conditions , . Indeed, if , then , and if , then . ◻

Consideration of the adjoint of , together with the reflexivity of , shows that we may also find an invertible for which is a projection. In case itself is invertible, we must have .

EXERCISES

Exercise 1. What is the rank of the differentiation operator on ? What is its nullity?

Exercise 2. Find the ranks of the following matrices.

Exercise 3. If is left multiplication by on a space of linear transformations (cf. Section: Matrices of transformations , Ex. 5), and if has rank , what is the rank of ?

Exercise 4. The rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks.

Exercise 5.

If and are linear transformations on an -dimensional vector space, and if , then .
For each linear transformation on an -dimensional vector space there exists a linear transformation such that and such that .

Exercise 6. If , , and are linear transformations on a finite-dimensional vector space, then

Exercise 7. Prove that two linear transformations (on the same finite-dimensional vector space) are equivalent if and only if they have the same rank.

Exercise 8.

Suppose that and are linear transformations (on the same finite-dimensional vector space) such that and . Is it true that and are similar if and only if ?
Suppose that and are linear transformations (on the same finite-dimensional vector space) such that , , and . Is it true that and are similar if and only if ?

Exercise 9.

If is a linear transformation of rank one, then there exists a unique scalar such that .
If , then is invertible.

EXERCISES

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