**Answer the following:**3. State why ⟨u, v⟩ is not an inner product for $ u = (u_1, u_2) $ and $ v = (v_1, v_2) $ in $ \mathbb{R}^2 $.\[ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \]---### ExplanationThis problem asks to verify why the given operation is not an inner product in $ \mathbb{R}^2 $. An inner product must satisfy four properties:1. **Conjugate Symmetry:** $ \langle u, v \rangle = \overline{\langle v, u \rangle} $2. **Linearity in the First Argument:** $ \langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle $ for all scalars $ a, b $ and vectors $ u, w $3. **Positive Definiteness:** $ \langle u, u \rangle \geq 0 $ with equality if and only if $ u = 0 $4. **Additivity in the First Argument:** $ \langle u+w, v \rangle = \langle u, v \rangle + \langle w, v \rangle $The given operation $ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 $ does not satisfy all of these conditions, most likely the positive definiteness or linearity, making it not a valid inner product.

We are given that u,v=u1v12+u2v22 for u=u1,u2 and v=v1,v2 in R2. We know that an inner product must…

Answered: State why (u, v) is not an inner…

Math

Advanced Math

State why (u, v) is not an inner product for u = (ui, u:) and v = (v, V2) in R? %3D (u, v) = (u,v;)² + (u2v2)²

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: ) Find the vector that parallel to the intersection line of two planes 2x +y-3z 2 and -x+2y-z 1.

A: Given problem:- Find the vector that parallel to the intersection line of two planes 2x + y - 3z = 2…

Q: m) Linearize : -3x 天ェ_= around %31

Q: Which of the following cannot be defined as an inner product between two vectors (x0, Yo) and (¤1,…

Q: Show that (u×v)· w = u ·(v×w).

A: As we know that, Multiplication is an Associative Property, i.e. (a x b).c=a.(b x c) and Sometimes…

Q: Consider the following vectors. 6, us Apply Gram-Schmidt to find an orthogonal basis for S. (a) S =…

A: a) S=spanu1,u2 orthogonal basis for s=w1,w2w1=u1=4-10w2=u2-u2,w1w1,w1w1 =381-12-8+016+1+04-10…

Q: Demonstrate the following relationship involving the dot product and cross product (a) u. (v x W): =…

Q: 8. If it [13,7.-2] is orthogonal to v v=[-5, k.7], determine the value(s) of k.

A: Given query is to find the value of k.n

Q: Let u = (3, 2, x) and v = (2x, 4, x). (a) Find all possible values of x that make u and v…

A: Given thata)We have to find all possible values of x that make u and v orthogonal.b)We have to find…

Q: If A is diagonalizable through D= (P^-1) A P then the diagonal factorization of A = P D (P^-1)…

Q: Find the projection of u = onto v = .

Q: Determine Cif u=v=0 (20 — 5и + 3)dv — (20 + 4и — 6)du %3D 0

Q: 22. Reflect in the x-axis. D 4+ 3- -6 -5 -4 -3 -2 F2 -2 4- -6- a. D'(-3, 4), E'(-6, 4), F'(-1, 0) b.…

A: When a coordinate is reflected about the x-axis then its y coordinate remains the same but the sign…

Q: Q/Find the real values of a , b, c that make the convolution equal to zero for the vector function…

Q: Which of the following are true for A = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 4), (3, 2),…

A: The given set .The given statement .The objective is to determine whether the statement is true or…

Q: Calculate the cross product assuming that u x w = (6, 3, 6) (-u+3w) x w =

Q: Express the area of a square (A) in terms of its diagonal (d)

Q: Find the angle in degrees between u = (1, 2, 0) and v = (1, -2, 1)

A: Given, u=1,2,0v=1,-2,1 The formula for finding the angle between the two vector is, cosθ=u→·v→u→v→

Q: Compute u X v.

A: Cross product of two vectors a x b is defined as a vector c, which is perpendicular to both the…

Q: Find the projection of u onto v if u= (5,-3), v = (-1,4).

Q: Vectors a' and b are perpendicular to one another. Find k : a'(k², –k, 1), b´ (k, 3, – 2)

A: Scalar product of two vectors

Q: H is the set of all vectors x = X2 in R³ with the property that X3 X1 + x2 = 0 and Xi – x2 + x3 = 0.…

Q: 24 Resolveu = [8, 3, 5] into rectangular components, one of which is collinear with v = [3, 1, 2].

Q: Q7) True or False, and Justify your answer. If v1, v2, ... V4 are in Rª and v3 = 2v1 + v2, then…

Q: If u • vx w or v • or w • v equal to zero, then the vectors u, v , and w are coplanar... True False

A: Concept Used: Three vectors are said to coplanar if their scalar triple product is zero That is [u→…

Q: Compute the dot product (7, 6, 8) - (4, 10, 8).

A: To find - Compute the dot product 7, 6, 8·4, 10, 8

Question

100%

**Answer the following:**

3. State why ⟨u, v⟩ is not an inner product for $ u = (u_1, u_2) $ and $ v = (v_1, v_2) $ in $ \mathbb{R}^2 $.

\[ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 \]

---

### Explanation

This problem asks to verify why the given operation is not an inner product in $ \mathbb{R}^2 $. An inner product must satisfy four properties:

1. **Conjugate Symmetry:** $ \langle u, v \rangle = \overline{\langle v, u \rangle} $
2. **Linearity in the First Argument:** $ \langle au + bw, v \rangle = a\langle u, v \rangle + b\langle w, v \rangle $ for all scalars $ a, b $ and vectors $ u, w $
3. **Positive Definiteness:** $ \langle u, u \rangle \geq 0 $ with equality if and only if $ u = 0 $
4. **Additivity in the First Argument:** $ \langle u+w, v \rangle = \langle u, v \rangle + \langle w, v \rangle $

The given operation $ \langle u, v \rangle = (u_1 v_1)^2 + (u_2 v_2)^2 $ does not satisfy all of these conditions, most likely the positive definiteness or linearity, making it not a valid inner product.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Q/Find the real values of a, b, c that make the convolution equal to zero for the vector function F(x, y, z) = (2az + y – 5x)i + (2x – z + 2cy)k + (y + 2bx + 3z)j
1. (2D5 - 7D* + 2D³ + 3D²)(f) = 0
Calculate the cross product assuming that u X w = (-2, -3, -2) (2u + 4w) x w = (
Calculate and simplify the wedge product A. = 3dx1 ^ dx 2 + 9dx₁ / dx4, =-9dx2 dx3 + 5dx3 ^ dx4. ΦΛΕ dx1 Adx2 Adr³ \\ dx4.
If F(u,v) = (u-v) / (u+v), Find F + F (u, v)
Find two units rectors orthogonal to both = {1,2,-4) and 22² = (0,5,8) 6