Answered: 1. Suppose V is the vector space of… | bartleby

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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Q1(a)

1. Suppose V is the vector space of continuous functions f: [-1, 1] → R.
(a Show that (,): V × V → R given by
defines an inner product.
1
(f,g) = [', f(x)g(a)(1 − a²) da
-1

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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 F and G be vector-valued functions such that ₹(t) = (cos(ät), e²t−1, t² − 1), Ĝ(1) = (1,1,−1), Ġ'(1) = (2, 3, 2), Evaluate (FG) (1).
Determine the interval (s) on which the vector-valued function r(t)=ti+(sqrtt+1j)+(t^2+1)k continous A [-infinity, -1) B [-infinity, infinity) C (-1, infinity) D [-1, infinity)
5. Let g(u) be a differentiable function, and let f(x, y) = g(x2 + y²). (a) Show that yfa = x fy. (b) Find the direction of maximal increase of f at the point (1,1) in terms of g'. (You do not need to provide this as a unit vector.)
Let f(w)f(w)be a function of vector ww, i.e. f(w)=1/(1+e−wTx). Determine the first derivative and matrix of second derivatives of ffwith respect to w ?
Let F and G be vector-valued functions such that F(t) = (cos(nt), e²t-1, t² – 1), Ġ(1) = (1, 1, – 1), Ğ'(1) = (2,3, 2), Ġ"(1) = (0,1,0) i. Find a vector equation of the tangent line to the graph of F at (-1, e,0). ii. Evaluate (F · G)'(1). iii. Evaluate (G x G')'(1). d'(1) = (2,3, 2), Ğ"(1) = (0,1,0) %3D

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