Let (a,b) and (c,d) be two vectors in R^2. If ad?bc=0, show that they are linearly dependent. If ad?bc ? 0, show that they are linearly independent.



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Let (a,b) and (c,d) be two vectors in R^2. If ad?bc=0, show that they are linearly...
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a) Show that A is symmetric (b) Show that N(A) S (c) Show that the rank of A must be 2.
Exercise 5 Let z and y be linearly independent vectors in R" and let S- span(,y). We can use r and y to define a matrix A by setting (a)...
Let M1, M2 and M3 are linearly independent sets of vectors, and that M, C M, CM2. Let M = M, UM, U M3. Then, which of the following is always true? a) M is linearly dependent b) M is linearly independent c) M is both linearly dependent and linearly independent d) M is neither linearly dependent nor linearly independent
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the x-y plane for the case ad - bc > 0. (a) Sketch D in the z-y plane for the case ad-bc 〈 0. (c) Calculate the area of D. Show all working.
(7) Let 0くa 〈 b 〈 c 〈...
Question 2. Let f(x) = azt for a, b, c, d e C, ad - bc +0. [2]a) Show lim + f() = oo if c = 0. [2]b) With c#0, show lim + f() = ? and lim,--4f(x) = 0.
Let u = and v= Determine whether the vectors u and v are linearly independent or linearly dependent, and choose the most correct answer below. A. The vectors are linearly independent. B. We cannot easily tell whether the vectors are linearly independent or linearly dependent. C. The vectors are linearly dependent.
(1 point) Suppose S = {r, u, d} is a set of linearly independent vectors. If x = 4r + 2u + 5d, determine whether T = {r, u, 2} is a linearly independent set. Select an Answer 1. Is T linearly independent or dependent? IfT is dependent, enter a non-trivial linear relation below. Otherwise, enter O's for the coefficients. u+ !!! I=0
Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation x=au + bu, y=cu + dv. (a) Sketch D in the r-y plane for the case ad -bc > 0. (a) Sketch D in the r-y plane for the case ad bc < 0. (c) Calculate the area of D. Show all working.
Let 0
(7) Let 0 < a < b < c 〈 d for a,b,c,de R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation ( d -bc > 0. a) Sketch D in the x-y plane for the case a -bc< (a) Sketch D in the r-y plane for the case ad 0. (c) Calculate the area of D. Show all working.
(7) Let 0
Let v1 and v2 be two arbitrary linearly independent vectors in R^n . Are (v1 + v2) and (v1 − v2) necessarily linearly independent? Justify.
10. Mobius transformations. Let
a, b, c, d ad-bc 0 . The
function is called a Mobius transformation (or linear fractional
transformation). Show that
a) lim z->inf T(z) = inf if c=0;
b)kim z-> inf T(z) = a/c and lim z-> d/c T(z) = inf if
c0
*10. Möbius transformations. Let a,b,c,d EC with ad-bc70. The function T(2) = 2 a2 + b cz + d à (2 +-d/c) is called a Möbius transformation (or linear fractional transformation). Show that...