Need Help?

  • Notes
  • Comments & Questions

Exercise 4 Determine the dimension of each of the following subspaces:
a. $$U_{7}=\left\{\left(\begin{array}{l}{x} \\ {y}\end{array}\right) | x-2 y=0\right\}$$

$$x-2 y=0$$

Let $$y=t \Rightarrow x=2 t$$

$$\left(\begin{array}{l}{x} \\ {y}\end{array}\right) \Rightarrow\left(\begin{array}{c}{2 t} \\ {t}\end{array}\right) \Rightarrow\left(\begin{array}{l}{2} \\ {1}\end{array}\right) t$$

$$U_{7}=Span\left(\begin{array}{l}{2} \\ {1}\end{array}\right)$$

$$dim\left(U_{7}\right)=1$$

b. $$U_{8}=\left\{\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) | x-2 y-z=0\right\} $$

$$x-2 y-z=0$$

Let $$y=r$$
Let $$z=s$$

$$x=2 r+s$$

$$\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) \Rightarrow\left(\begin{array}{c}{2 r+s} \\ {r} \\ {s}\end{array}\right)=\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right) r+\left(\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right) s$$

$$U_{8}=span\left(\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right),\left(\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right)\right)$$

$$dim(U_{8})=2$$

c. $$U_{9}=\left\{\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) | x-2 y=0\right\} $$

$$x-2 y=0$$

Let $$y=r$$
Let $$z=s$$

$$x=2 r$$

$$\left(\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right) \Rightarrow\left(\begin{array}{l}{2 r} \\ {r} \\ {s}\end{array}\right)=\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right) r+\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right) s$$

$$U_{9}=span\left(\left(\begin{array}{l}{2} \\ {1} \\ {0}\end{array}\right),\left(\begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right)\right)$$

$$\Rightarrow dim\left(U_{1}\right)=2$$

No comments yet

Join the conversation

Join Notatee Today!