Assume that f: Fn-+ Fn, X- А.x is bijective. Show that A E Fnxn is invertible....
injective D Question 22 [Extra Credit] Let f:(-2,2)=(0,4); f(x)=x? Is f invertible? No. f is neither 1-1 nor onto. Yes, because fis bijective. No. While fis onto, it is not 1-1. No. While fis 1-1, it is not onto.
(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form A 0 ( BC where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of the...
(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where 0 is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form (Α (в ο). с) where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of...
「 : / (2) Let A- be an arbitrary 2 x 2 matrix. (a) If A is invertible, perform row operations to determine a row echelon form of A. (Hint: You may need to consider different cases, e.g., when a-0 and when a f 0.) (b) Under certain conditions, we can row reduce [A | 2 to [| B] where d -b ad- be-a Use the row echelon form of A from part (a) to find conditions under which the...
part (c) 7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
5. Let f(x) = arctan(In x) for all x >0. A graph of y = f(x) is shown in the figure. (a) Find the formula for the derivative f'(x). Then explain how you can deduce from this formula that f is invertible. (b) Find the formula for f-1(x), the inverse of f. (c) What is the domain and range of f-1? (d) Sketch a graph of the function y=f-1(x). (e) Now determine the value of (F-1)(0) using your results from...
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
Let f(x)-1 if XE [-1/2, 1 /2) x 2 if xE[1/2, T] If FN(x) is the partial sum of the Fourier series of f(x), then give lim Fv(1/2)? Please give your answer in decimal form. (Hint: It might be helpful to sketch the function) Let f(x)-1 if XE [-1/2, 1 /2) x 2 if xE[1/2, T] If FN(x) is the partial sum of the Fourier series of f(x), then give lim Fv(1/2)? Please give your answer in decimal form. (Hint:...
[1 2 37 1. Is the matrix 1 0 1 invertible? If yes, what is its inverse? [O 2 -1 2. A matrix is called symmetric if At = A. What can you say about the shape of a symmetric matrix? Give an example of a symmetric matrix that is not a zero matrix. 3. A matrix is called anti-symmetric if A= -A. What can you say about the shape of an anti- symmetric matrix? Give an example of an...
3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform 3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform