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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

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Question

Let Y = R([a, b]) be the set (actually a vector space) of bounded,
real-valued, Riemann integrable functions on the closed and bounded
interval [a, b], and let X = C([a, b]), be the subset (actually a vector
subspace) of continuous, real-valued functions on [a, b]. For f, g = Y,
define
d(f,g) = sup |f(x) — g(x)|.
x= [a,b]
Does every uniformly Cauchy sequence in R([a, b]) converge uni-
formly to a function in R([a, b])?

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Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
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 V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.
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.
Prove that in a given vector space V, the zero vector is unique.
Consider the vector space of continuous functions over the interval [-1,1] with the inner product Then the cosine of the angle between f(x)= 3x² and g(x)=2x3 is
1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2]. Let f, g E V and define the inner product of f and g to be (f, g) = | f(x)g(x) dx. Let h(x) = x? and let k(x) = 7x*, compute (h, k). Simplify your answer completely.
Let V be the set of ordered pairs (x, y) of real numbers and let F be the field of real numbers. Define (x. v)+ (x,: V2)= (x + x,, 0) c(x1. Vi)=(cx. 0). Examine whether V is a vector space or not ? If not, list the properties which fail to hold.
3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition + : V × V → V defined f: via (f+g)(x) = f(x) + g(x), x = R", and scalar multiplication : Rx V → V defined via (A. f)(x) = \f(x), ● x ER". Now let W = {f: R" → R: f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R" → R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). (b) Find a basis for W. You should prove that it is indeed a basis.
Please work out
Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1 + x2 + 1, y1 + Y2+ 1) and k (*1, Y1) = (kx1, kyı) is not a vector space.

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