$$\begin{array} { l } { \text { Prove that isomorphism is an equivalence relation. That is, for any } } \\ { \text { groups } G , H , \text { and } K , G \approx G , G \approx H \text { implies } H \approx G , \text { and } G \approx H \text { and } } \\ { H \approx K \text { implies } G \approx K } \end{array}$$

A QR factorization of A is given. Use it to find a least squares solution of

$$A \mathbf { x } = \mathbf { b }.$$

$$A = \left[ \begin{array} { l l } { 2 } & { 1 } \\ { 2 } & { 0 } \\ { 1 } & { 1 } \end{array} \right] , Q = \left[ \begin{array} { r r } { \frac { 2 } { 3 } } & { \frac { 1 } { 3 } } \\ { \frac { 2 } { 3 } } & { - \frac { 2 } { 3 } } \\ { \frac { 1 } { 3 } } & { \frac { 2 } { 3 } } \end{array} \right] , R = \left[ \begin{array} { l l } { 3 } & { 1 } \\ { 0 } & { 1 } \end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } { 2 } \\ { 3 } \\ { - 1 } \end{array} \right]$$

$$\text { The letters of the alphabet can be sorted into the following }\begin{array} { l } { \text { categories: } } \\ { \text { 1. FGJLNPQRSZ } } \\ { \text { 2. BCDEK } } \\ { \text { 3. AMTUVWY } } \\ { \text { 4. HIOX } } \\ { \text { What defines the categories? } } \end{array}$$