Answered: Create 2 sets, one linearly independent…

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

Concept explainers

Question

Create 2 sets, one linearly independent and one linearly dependent, by replacing the x with a number, starting from B = { (1,2,-1,8), (-2,-1,4,2), (13,14,-21,x) }

Expert Solution

Step 1

Step: 1

Linearly Dependent: - A set of vectors is called linearly dependent if at least one element of the set can be written as linear combination of others elements.

Linearly Dependent: - If no vector (element) of set can be written as linear combination of others elements. Then set is called linearly dependent.

Step: 2

Now, given that $B = {(1, 2, - 1, 8), (- 2, - 1, 4, 2), (13, 14, - 21, x)}$

Let B is linearly dependent set then at least one element of B can be written as linear combination of others element.

$S o, (13, 14, - 21, x) = a . (1, 2, - 1, 8) + b . (- 2, - 1, 4, 2) H e r e, a a n d b a r e s c a l a r .$

comparing both sides we get,

$a - 2 b = 13 - - (1) 2 a - b = 14 - - (2) - a + 4 b = - 21 - - (3) - a + 2 b = x - - (4) s o l v i n g e q u a t i o n n (1) a n d (2), t o f i n d v a l u e o f a a n d b f r o m (1), a = 13 + 2 b p u t i n (2), 2 a - b = 14 2 (13 + 2 b) - b = 14 26 + 4 b - b = 14 3 b = - 12 b = - 4$

$w e h a v e, a = 13 + 2 b a = 13 + 2 \times (- 4) a = 13 - 8 a = 5$

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

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