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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q22E (page 261)

Iff(t)is a continuous function, what is the relationship between
${鈭�}_{0}^{1} {(f (t))}^{2} d t and {({鈭�}_{0}^{1} f (t) dt)}^{2}$
Hint: Use the Cauchy鈥揝chwarz inequality.

Short Answer

Expert verified

${({鈭�}_{0}^{1} f (t) d t)}^{2} 鈮� {鈭�}_{0}^{1} {(f (t))}^{2} d t$

Step by step solution

01

Cauchy–Schwarz inequality.

If $\overset{鈫�}{x}$ and $\overset{鈫�}{y}$ are vectors in $R^{n}$ then $| \overset{鈫�}{x}, \overset{鈫�}{y} | 鈮� || \overset{鈫�}{x} || . || \overset{鈫�}{y} ||$ .

This statement is an equality if (and only if) and are parallel.

It will apply theCauchy鈥揝chwarz inequality.

By taking the functions f and 1.

Observe that following,

${({鈭�}_{0}^{1} f (t) d t)}^{2} = {|<f, 1>|}^{2} = 鈮� {||f||}^{2} . ||1|| = {鈭�}_{0}^{1} {(f (t))}^{2} d t$

Hence, the relationship between ${鈭�}_{0}^{1} {(f (t))}^{2} d t and {({鈭�}_{0}^{1} f (t) d t)}^{2} is {({鈭�}_{0}^{1} f (t) d t)}^{2} 鈮� {鈭�}_{0}^{1} {(f (t))}^{2} d t$ .

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Most popular questions from this chapter

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof $A^{T}$ .

TRUE OR FALSE?If $\overset{鈬赌}{x} a n d \overset{鈬赌}{y}$ are two vectors in $R^{n}$ , then the equation role="math" localid="1659506190737" $鈭� \overset{鈫�}{x} + \overset{鈫�}{y} {鈭�}^{2} = 鈭� \overset{鈫�}{x} {鈭�}^{2} + 鈭� \overset{鈫�}{y} {鈭�}^{2}$ must hold.

Consider a QRfactorization

M=QRShow that $R = Q^{T} M$ .

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

20. $[\begin{matrix} 2 & 3 & 5 \\ 0 & 4 & 6 \\ 0 & 0 & 7 \end{matrix}]$

If A is an $n 脳 n$ matrix such that role="math" localid="1659514225617" ${AA}^{T} = I_{n}$ , then A must be an orthogonal matrix.

See all solutions

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