Let (X, d) be a metric space. Define f: Xx X → R by f(x, y) = = that f is a metric on X. d(x, y) 1+ d(x, y) Prove
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- Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].Let z = f(x, y) be a function whose graph passes through the point (a, b, c), and consider its graph as the level surface of w = g(x, y, z) = f(x,y) – z at the level w = 0. There is a unit vector ú that is both tangent to the surface = f(x,y) at (a, b, c) and parallel to the gradient vector Vg(a, b, c). O always sometimes O never
- let (x,d) be a metric space. define a flow on (x,d)Q4) Consider the space (X.T) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).Let d: R2 × R2 →R given by xỉ + yỉ + x3 + y½ , if (x1,y1) # (x2, Y2) 0, if (x1, Y1) = (x2, Y2) d((x1.y1).(x2.y2))= { (a) Show that d is a metric. (b) Show that (R2, d) is a complete metric space. (c) Show that (R2, d no a connected metric space
- Tutorial 1 (1.1) Let B = {r, y} and c> 0. Suppose the function f on B × B satisfies Sf(2, 2) = f(y, y) = 0, f(r, y) = f(y, ) = c. Show that f is a metric on B.Consider inner-product = | f(x)g(x) dx defined for vector space C[-1, 1] , -1 then for the function f(x) = 3 x and g(x)=-2 the inner-product equals: Select one: a. zero b. 1 C. -6 d. 6Prove that span( { x}) = {ax: a ϵ F} for any vector x in a vedor space. Interpret this result geometrically in R3 .
- What is the dimension of the vector space R 5Check whether or not the set V is vector space over R2 with respect to the indicated operations. ( x1, y1 ) + ( x2, y2 ) = ( x1 - x2, y1- y2 ) k ( x1, y1 ) = (-kx1, -ky1 )Please use similar notation from the images, hope they help, thank you. Let V be an n-dimensional vector space. Define f : V \rightarrow \R by f(v) = |v2|. Let p,q \in V. Find dfp(q).