Linear Algebra with Applications, 10th Edition

5.2

1. 1. 

   1. (a) $\{{(3,4)}^T\}$ basis for $R(A^T)$,

      $\{{(-4,3)}^T\}$ basis for N(A),

      $\{{(1,2)}^T\}$ basis for R(A),

      $\{{(-2,1)}^T\}$ basis for $N(A^T)$;

   2. (d) basis for $R(A^T)$:

      $\{{(1,0,0,0)}^T,{(0,1,0,0)}^T{(0,0,1,1)}^T\}$,

      basis for N(A): $\{{(0,0,-1,1)}^T\}$,

      basis for R(A): $\{(1,0,0,1{)}^T,{(0,1,0,1)}^T{(0,0,1,1)}^T\}$,

      basis for $N(A^T)$: $\{(1,1,1,-1{)}^T\}$

2. 2. 

   1. (a) Hint: A vector y is in $S^{\perp }$ if and only if ${\mathbf{y}}^T\mathbf{\text{x}}=\; 0$.

3. 3. 

   1. (a) Hint: A vector z will be in $S^{\perp }$ if and only if z is orthogonal to both x and y.

   2. (b) The orthogonal complement is spanned by $(-5,1,3{)}^T$.

4. 4. $\{{(-1,2,0,1)}^T,(2,-3,1,0{)}^T\}$ is one basis for $S^{\perp }$.

5. 6. Hint: If (3, 1, 2) is in the row space of A, then ${(3,1,2)}^T$ is in the column space of $A^T$.

6. 7. Hint: What do we know about the intersection of $N\left(A^T\right)$ and R(A)?

7. 10. Hint: If $A\mathbf{\text{x = b}}$ has no solution, then $\mathbf{\text{b}}\notin R$