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Problem 2.8(ii)) Consider the operatorsR2R2 and U R?R2 wherealdpoint) Prove that 0 adl & are linear mapS . Are and & composites of rotations #d rellections ...

Question

Problem 2.8(ii)) Consider the operatorsR2R2 and U R?R2 wherealdpoint) Prove that 0 adl & are linear mapS . Are and & composites of rotations #d rellections in R?? point ) Compute (the formulas for) 02 = 00 0 ad 42 = 0o 0 points) Compute @ 0 and (0 0 Vo 0-

Problem 2.8(ii)) Consider the operators R2 R2 and U R? R2 where ald point) Prove that 0 adl & are linear mapS . Are and & composites of rotations #d rellections in R?? point ) Compute (the formulas for) 02 = 00 0 ad 42 = 0o 0 points) Compute @ 0 and (0 0 Vo 0-



Answers

Determine whether the operators $T_{1}$ and $T_{2}$ commute; that is, whether $T_{1} \circ T_{2}=T_{2} \circ T_{1}$. (a) $T_{1}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $x$ -axis, and $T_{2}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $y$ -axis. (b) $T_{1}: R^{2} \rightarrow R^{2}$ is the rotation about the origin through an angle of $\pi / 4,$ and $T_{2}: R^{2} \rightarrow R^{2}$ is the reflection about the $y$ -axis.


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