Answered: Show that (u₁, 4₂) is an orthogonal…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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	2
−8

	16
4

	9
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Question content area bottom

Part 1

Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for

ℝ2?

Select all that apply.

The vectors must all have a length of 1.

The vectors must span

ℝ2.

Your answer is correct.

The vectors must form an orthogonal set.

Your answer is correct.

The distance between any pair of distinct vectors must be constant.

Part 2

Which theorem could help prove one of these criteria from another?

S=u1, ..., up

and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.

S=u1, ..., up

and each

has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.

S=u1, ..., up

is an orthogonal set of nonzero vectors in

ℝn,

then S is linearly independent and hence is a basis for the subspace spanned by S.

Your answer is correct.

S=u1, ..., up

is a basis in

ℝp,

then the members of S span

ℝp

and hence form an orthogonal set.

Part 3

What calculation shows that

u1, u2

is an orthogonal basis for

ℝ2?

Since the

inner product of Bold u 1 and Bold u 2inner product of u1 and u2

00,

the vectors

form an orthogonal set.

From the theorem above, this proves that the vectors are also

a basis for set of real numbers R squareda basis for ℝ2

because they are two

linearly independent vectors in set of real numbers R squared .linearly independent vectors in ℝ2.

Linear Algebra

Part 4

Express x as a linear combination of the

u's.

x=enter your response hereu1+enter your response hereu2

(Simplify your answers. Use integers or fractions for any numbers in the equation.)

Show that {₁, ₂} is an orthogonal basis for R². Then express x as a linear combination of the u's.
2
U₁ =
, U₂
- 8
16
4
9
- 2
vynich theorem couid neip prove one of these crea from another?
A. If S = {₁,
{U₁,
C. If S= = {U₁,
D. If S= = {U₁,
X =
=
B. If S=
and x =
up and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
up} and each u¡ has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
up} is an orthogonal set of nonzero vectors in R", then S is linearly independent and hence is a basis for the subspace spanned by S.
up} is a basis in RP, then the members of S span RP and hence form an orthogonal set.
What calculation shows that (u₁, u₂} is an orthogonal basis for R²?
Since the inner product of u₁ and ₂ is 0, the vectors form an orthogonal set. From the theorem above, this proves that the vectors are also
linearly independent vectors in R².
Express x as a linear combination of the u's.
= μ₁ + ¹₂
(Simplify your answers. Use integers or fractions for any numbers in the equation.)
a basis for R²
because they are two

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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