.
What is AT? What is A2? What is ATA? (Do these computations both by hand and with numpy.)
2. The notation ||x||2 means the Euclidean norm (also called the 2-norm, or just the length) of the vector x. What is ||(3,1,4,1,5)T||2?
3. Consider the following system of three equations in three unknowns.
2x0 − 3x1 + x2 = 1 (1)
2x1 + 3x2 = 7 (2)
x0 + x2 = 4 (3)
First write this system in the form Ax = b, where A is a matrix and x and b are vectors. Second, write two lines of numpy code that use np.array() to create A and b as numpy arrays.
4. What vector x solves the system above?
5. Write down a 2-by-2 matrix A and a 2-vector b such that Ax = b has no solution. Explain in a sentence why there are no solutions.
6. Write down a 2-by-2 matrix A and a 2-vector b such that Ax = b has more than one solution. Give two different solutions.
7. Is there a 2-by-2 matrix A and a 2-vector b such that Ax = b has exactly two solutions? Why or why not?
8. Recall that a number λ is an eigenvalue of a matrix A if there is some vector x (an eigenvector) for which Ax = λx. Give an eigenvalue of the following matrix, and a corresponding eigenvector.
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9. If f(x) = 7x3 − 2x2 + 4x − 5, what is f′(x), the derivative of f(x)?
10. If z = xey/2, what is ∂z/∂x? What is ∂z/∂y?
11. If f′(x) = x2 + sinx, what is f(x)?
12. The height in feet of a projectile fired straight up is given by h = 1280t−16t2, where t is in seconds. What is the maximum height the projectile will reach? When will it hit the ground?
13. Let f(x,y) = x2 − xy + 3y2. We see that f(3,2) = 15. In which direction from the point (3,2) does f(x,y) decrease the fastest? (The direction is a vector in the (x,y) plane.)
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