We returned to the topic of the SVD, and we finally completed the proof of its existence. After that, we proved two short results.

# Subsection20.1Exercises

##### Exercise20.3

We proved in class that the SVD of any matrix exists. Carefully read the details in the Lecture 4 of the textbook that demonstrate that, if the matrix is square and the singular values distinct, then the left and right singular vectors are uniquely determined up to complex sign.

##### Exercise20.4

In class we proved that $\range(A)=\Span(u_1, u_2, \dots, u_r),$ where $r$ is the number of nonzero singular values of $A\text{.}$ Now prove that

\begin{equation*} \Null(A)=\Span(v_{r+1},\dots, v_n). \end{equation*}