Exercise 7.D.14

Proof. First, we will prove that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}T$. From Exercise 5 in section 7A we see that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}$.

Suppose $w\in \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$. Then $w\in \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}$. Thus $w=T^{\text{∗}}v$ for some $v\in V$. We can write $v=v^{\prime }+v^{\mathit{\prime \prime }}$ for some $v^{\prime }\in \mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}$ and $v^{\mathit{\prime \prime }}\in {(\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}})}^{\text{⊥}}$. Thus $w=T^{\text{∗}}{v}^{\mathit{\prime \prime }}$. But 7.7 shows that ${(\mathrm{null}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}})}^{\text{⊥}}=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$. Therefore $v^{\mathit{\prime \prime }}=\mathit{Tu}$ for some $u\in V$ and so $w=T^{\text{∗}}\mathit{Tu}\in \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}T$. Hence $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T\subset \mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}T$.

The inclusion in the other direction is easy. We have

Therefore $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}T$.

Since $\sqrt{T^{\ast }T}$ is diagonalizable (because it is self-adjoint), it follows that the number of nonzero singular values of $T$ equals the dimension of $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->\sqrt{T^{\ast }T}$. Note that $\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T^{\text{∗}}T=\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->\sqrt{T^{\ast }T}$. Therefore $\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->\sqrt{T^{\ast }T}=\dim <!--\; FUNCTION\; APPLICATION\; -->\mathrm{range}<!--\; FUNCTION\; APPLICATION\; -->T$, completing the proof. □