1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
4vir4ik [10]
3 years ago
5

If vectors u,v and w ar linearly indepndnt, will u-v,v-w & u-w also be linearly indpnt? ...?

Mathematics
1 answer:
inna [77]3 years ago
6 0
No they won’t be.Consider the linear combination (1)(u – v) + (1) (v – w) + (-1)(u – w).This will add to 0. But the coefficients aren’t all 0.Therefore, those vectors aren’t linearly independent.
You can try an example of this with (1, 0, 0), (0, 1, 0), and (0, 0, 1), the usual basis vectors of R3.
That method relied on spotting the solution immediately.If you couldn’t see that, then there’s another approach to the problem.
We know that u, v, w are linearly independent vectors.So if au + bv + cw = 0, then a, b, and c are all 0 by definition.
Suppose we wanted to ask whether u – v, v – w, and u – w are linearly independent.Then we’d like to see if there are non-zero coefficients in the linear combinationd(u – v) + e(v – w) + f(u – w) = 0, where d, e, and f are scalars.
Distributing, we get du – dv + ev – ew + fu – fw = 0.Then regrouping by vector: (d + f)u + (-d +e)v + (-e – f)w = 0.
But now we have a linear combo of u, v, and w vectors.Therefore, all the coefficients must be 0.So d + f = 0, -d + e = 0, and –e – f = 0. It turns out that there’s a free variable in this solution.Say you let d be the free variable.Then we see f = -d and e = d.
Then any solution of the form (d, e, f) = (d, d, -d) will make (d + f)u + (-d +e)v + (-e – f)w = 0 a true statement.
Let d = 1 and you get our original solution. You can let d = 2, 3, or anything if you want.

You might be interested in
What is the best description of parallel lines?
hichkok12 [17]
1,3,5,6 are correct answers
3 0
2 years ago
I really hate to do this again-
babymother [125]

Answer:

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
Find the measure of an angle between 0 and 360 that is coterminal with an angle measuring -800
sertanlavr [38]

Answer

280

Step-by-step explanation:

3 0
3 years ago
Read 2 more answers
The first diagram shows an equilateral triangle and a
liberstina [14]

Answer:

Ar = ¼√3 As

Step-by-step explanation:

Area of an equilateral triangle is:

Ar = ¼√3 s²

Area of a square is:

As = s²

Substituting:

Ar = ¼√3 As

4 0
3 years ago
You are designing an aquarium that is to have a tank holding 225,000 gallons of water. The tank will be 100 feet long and 20 fee
vladimir2022 [97]
Well I'm just doing some random math so it might be 300?
5 0
3 years ago
Other questions:
  • the ages of students in a computer class are 14, 13, 14,15,14,35,14. Decide which measure of center(s) best describes the data s
    14·1 answer
  • A one year membership to a gym costs $725. The registration fee is $125, and the remaining amount is paid monthly. Defibr a vari
    14·2 answers
  • Round each number to the nearest 10 5,389
    12·2 answers
  • The front tent is 8 feet and 5 feet tall what is the part of the tent
    11·1 answer
  • What is the answer to this expression?
    6·1 answer
  • Oh dear god i don't even know what to do with this problem, please help me
    10·1 answer
  • Help me please!!!!!!
    7·1 answer
  • How to do this step by step
    8·2 answers
  • Daniel wants to do 312 practice problems to prepare for an exam. If he wants to do approximately the same number of problems eac
    11·1 answer
  • Kylie is selling gift bags and wrapping paper for a school fundraiser. Mrs. Jones orders 4 gift bags and 3 rolls of wrapping pap
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!