Could someone help me out with # 1 determine the points at which each function h
ID: 2966383 • Letter: C
Question
Could someone help me out with # 1
determine the points at which each function has a derivative, and wherever/ exists, find it. In Section 5 we saw that if the limit of a function/= u + iv exists, then the limit of each of its component functions u and v exists, and vice versa. Similarly, we saw that if a function is continuous, its components are continuous, and conversely. Give an example to illustrate the fact that this intimate relation between a function and its components fails in the case of differentiation. Using a careful definition, one can identify the function/z) = Arg z with the function /z) = arctan (y/x) for all z = x + iy where x > o. Assuming this identification, show that the function /z) = Arg z has no derivative anywhere. Suppose that /z) = u (x, y) + iv (x, y) is differentiable at a nonzero point. Then prove that at that point the polar form of the Cauchy-Riemann equations is r . ur = v theta and r . vr = -u theta.Explanation / Answer
We already know that (1), (2), and (3) are equivalent. The equivalence of (4) and (2) follows by the definition of the image. The equivalence of (4) and (5) follows because the only subspace of Rm that has dimension m is the whole space. Finally, the equivalence of (5) and (6) follows from the rank nullity theorem: since n=rank(A)+nullity(A), then nullity(A)=n?rank(A). So the rank equals m if and only if the nullity equals n?m. QED
So now you have a whole bunch of ways of checking if a matrix is one-to-one, and of checking if a matrix is onto. None of them is "better" than the others: for some matrices, one will be easier to check, for other matrices, it may be a different one which is easy to check. Also, the rank of a matrix is closely related to its row-echelon form, so that might help as well.
Note a few things: generally, "onto" and "one-to-one" are independent of one another. You can have a matrix be onto but not one-to-one; or be one-to-one but not onto; or be both; or be neither. The Rank-Nullity Theorem does place some restrictions: if A is m