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# HW7 - AMS210.01 Homework 7 Andrei Antonenko This homework...

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AMS210.01. Homework 7 Andrei Antonenko This homework is optional. You can solve it to get some extra-points in this class (contact me to get them). I recommend you to solve problems from it — this will really help you during your final! 1. Let the coordinates of the vector v in the “old” basis are v = (6 , 9 , 14). Find the coordinates of the vector in the “new” basis { e 1 = (1 , 1 , 1) , e 2 = (1 , 1 , 2) , e 3 = (1 , 2 , 3) } . 2. Let the “old” basis is { e 1 = (1 , 2 , 1) , e 2 = (2 , 3 , 3) , e 3 = (3 , 8 , 9) } , and “new” basis is { e 0 1 = (3 , 5 , 8) , e 0 2 = (5 , 14 , 13) , e 0 3 = (1 , 9 , 2) } . Find the change-of-basis matrix. 3. How does the change-of-basis matrix change if we (a) interchange two vectors of the “old” basis? (b) interchange two vectors of the “new” basis? (c) take vectors of both bases in reverse order? 4. Determine, which of the following mappings are linear operators. Justify your answer. (a) x 7→ a , where a is a fixed vector. (b) x 7→ x + a , where a is a fixed vector. (c) x 7→ αx , where α is a given number. (d) x 7→ h x, a i b , where a, b are fixed vectors, and V is a Euclidean space. (e) f ( x ) 7→ f ( x + 2), where f is a function. (f) f ( x ) 7→ f ( x + 1) - f ( x ), where f is a function. (g) ( x 1 , x 2 , x 3 ) 7→ ( x 1 + 2 , x 2 + 5 , x 3 ). (h) ( x 1 , x 2 , x 3 ) 7→ ( x 1 , x 2 , x 1 + x 2 + x 3 ). 5. Find the matrix of the following linear operators: (a) ( x 1 , x 2 , x 3 ) 7→ ( x 1 , x 1 + 2 x 2 , x 2 + 3 x 3 ) in the standard basis.

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