11.8. EXERCISES 209

Corollary 11.7.4 Let A be an m×n matrix. Then the rank of A and A∗equals the numberof singular values.

Proof:Since V and U are unitary, it follows that

rank(A) = rank(U∗AV ) = rank

(σ 00 0

)= number of singular values.

Also since U,V are unitary,

rank(A∗) = rank(V ∗A∗U) = rank((U∗AV )∗

)= rank

((σ 00 0

)∗)= number of singular values. ■

This is based on the simple observation that for A an m× n matrix, the dimension ofIm(A) is the same as the dimension of Im(UAV ) if U,V are invertible matrices of theright size. Indeed, Im(UAV ) = Im(UA) because V being invertible maps Fn onto Fn. Thedimension of Im(UA) and the dimension of Im(A) must be the same because U is one toone. Thus if a basis for Im(A) is {a1, · · · ,ak} , columns of A, then a basis for UA will be{Ua1, · · · ,Uak} .

11.8 Exercises1. Let {u1, · · · ,un} be a basis for Fn and define a mapping T : Fn→ span(v1, · · · ,vr)

as follows.

T

(n

∑k=1

akuk

)≡

r

∑k=1

akvk

Explain why this is a linear transformation.

2. In the above problem, suppose vk =uk. Show that Tv = v if v ∈V ≡ span(u1, · · · ,ur) .Now show that T (T (x)) = T (x) and that |Tx−Ty| ≤ |x−y| .

3. Find the minimum polynomials for the following matrices and use to obtain theeigenvalues of the matrix. The set of all eigenvectors associated with an eigenvalueλ is called the eigenspace. Determine the eigenspaces for each of these matrices.

(a)

(9 4−20 −9

)

(b)

(−3 −210 6

)

(c)

 5 −2 22 0 1−4 2 −1

