Let A be a matrix whose singular value decomposition is given by

$$

$$

$$

$$

$$

$$

$$

$$

$$

$2$

$52$

$5$

$2$

$52$

$5$

$3$

$5$

$2$

$5$

$2$

$5$

$3$

$5$

$2$

$5$

$2$

$5$

$2$

$52$

$5$

$2$

$5$

$3$

$52$

$5$

$3$

$52$

$5$

$2$

$52$

$52$

$5$

$2$

$5$

$3$

$5$

$2$

$52$

$52$

$5$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$40000$

$02000$

$00100$

$0000$

$0000$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$1$

$2$

$1$

$21$

$21$

$2$

$1$

$21$

$21$

$2$

$1$

$2$

$1$

$21$

$2$

$1$

$21$

$2$

$1$

$2$

$1$

$2$

$1$

$2$

$1$

$2$

$$

$$

$$

$$

$$

$$

a$.$ Determine the rank of A$.$

b$.$ Find an orthonormal basis for C$($A$).$

c$.$ Find an orthonormal basis for C$($AT $).$

d$.$ Find an orthonormal basis for N $($A$).$

e$.$ Find an orthonormal basis for N $($AT $).$