Answered: 3. (a) Prove that the standard inner… | bartleby

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 12EQ

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3. (a) Prove that the standard inner product on R²:
is an inner product.
(x, y) = x1y₁ + x2Y2,

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Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is an inner product in R2.
True or False: (p, q) = pq defines an inner product on P(R).
Calculate the cross product for (j − k) × (j + k)
Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p|, |la|l, and d(p, q) for the polynomials in P₂. p(x) = 1 -x + 4x², g(x) = x - x² (a) (p, q) (b) ||p|| (c) |||| (d) d(p, q)
Compute the dot product for (i + j) · (j + k)
Determine whether the function defined by = 2x1y1 − X1Y2 − X2y1 + 2x2y2 - - is an inner product on R².
. Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1) (x2,y2) = (4x1 + x2 − 4, Y1 + 3 y2 − 3) c(x,y) = (cx- c+1, cy − c + 1). (a) Show that scalar multiplication distributes over vector addition, that is: c((x1,y1)(x2,y2)) = (c○ (x1,y1)) + (c○ (x2,y2)). (b) Explain why V nonetheless is not a vector space by showing that a vector space property does not hold for this set with these operations.
Show that the function a Y1 (x, y) = [21 2] %3D Y2 is an inner product on R? if and only if b² – ac 0. Real analysis
Use the functions f and g in C[-1, 1] to find (f, g), f, g, and d(f, g) for the inner product (f, g) = f(x)g(x)dx. f(x) = 3x, g(x) = ex (a) (f, g) (b) ||f|| (c) ||g|| (d) d(f, g)
Show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3).
Let xbe inner Product and ||x1|=|ly|| = 1 x+y 1x-y ? why?
Xnt< why this is inner Product?

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