3. [10 points] For each isometry f: R³ R³, there is an induced map L₁: R³ R³ defined by (a,b,c) + f(a,b,c)-f(0,0,0). (Hi

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3 10 Points For Each Isometry F R R There Is An Induced Map L R R Defined By A B C F A B C F 0 0 0 Hi 1 (30.83 KiB) Viewed 35 times

3. [10 points] For each isometry f: R³ R³, there is an induced map L₁: R³ R³ defined by (a,b,c) + f(a,b,c)-f(0,0,0). (Hint: for all the parts below, it will probably help to draw a picture of what happens in a particular example, and to use geometric definitions of e.g. vector addition.) (a) Show that for any vector (x,y,z), we have L₁(a,b,c) = f(a+x,b+y,c+z)-f(x,y,z). (b) Show that L, is a linear map on R³ and is an isometry. (c) For each type of isometry of R³, say what type of isometry Lf is, with a brief justification.