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
Vikki [24]
3 years ago
13

Determine which of the sets of vectors is linearly independent. A: The set where p1(t) = 1, p2(t) = t2, p3(t) = 3 + 3t B: The se

t where p1(t) = t, p2(t) = t2, p3(t) = 2t + 3t2 C: The set where p1(t) = 1, p2(t) = t2, p3(t) = 3 + 3t + t2
Mathematics
1 answer:
defon3 years ago
4 0

Answer:

The set of vectors A and C are linearly independent.

Step-by-step explanation:

A set of vector is linearly independent if and only if the linear combination of these vector can only be equalised to zero only if all coefficients are zeroes. Let is evaluate each set algraically:

p_{1}(t) = 1, p_{2}(t)= t^{2} and p_{3}(t) = 3 + 3\cdot t:

\alpha_{1}\cdot p_{1}(t) + \alpha_{2}\cdot p_{2}(t) + \alpha_{3}\cdot p_{3}(t) = 0

\alpha_{1}\cdot 1 + \alpha_{2}\cdot t^{2} + \alpha_{3}\cdot (3 +3\cdot t) = 0

(\alpha_{1}+3\cdot \alpha_{3})\cdot 1 + \alpha_{2}\cdot t^{2} + \alpha_{3}\cdot t = 0

The following system of linear equations is obtained:

\alpha_{1} + 3\cdot \alpha_{3} = 0

\alpha_{2} = 0

\alpha_{3} = 0

Whose solution is \alpha_{1} = \alpha_{2} = \alpha_{3} = 0, which means that the set of vectors is linearly independent.

p_{1}(t) = t, p_{2}(t) = t^{2} and p_{3}(t) = 2\cdot t + 3\cdot t^{2}

\alpha_{1}\cdot p_{1}(t) + \alpha_{2}\cdot p_{2}(t) + \alpha_{3}\cdot p_{3}(t) = 0

\alpha_{1}\cdot t + \alpha_{2}\cdot t^{2} + \alpha_{3}\cdot (2\cdot t + 3\cdot t^{2})=0

(\alpha_{1}+2\cdot \alpha_{3})\cdot t + (\alpha_{2}+3\cdot \alpha_{3})\cdot t^{2} = 0

The following system of linear equations is obtained:

\alpha_{1}+2\cdot \alpha_{3} = 0

\alpha_{2}+3\cdot \alpha_{3} = 0

Since the number of variables is greater than the number of equations, let suppose that \alpha_{3} = k, where k\in\mathbb{R}. Then, the following relationships are consequently found:

\alpha_{1} = -2\cdot \alpha_{3}

\alpha_{1} = -2\cdot k

\alpha_{2}= -2\cdot \alpha_{3}

\alpha_{2} = -3\cdot k

It is evident that \alpha_{1} and \alpha_{2} are multiples of \alpha_{3}, which means that the set of vector are linearly dependent.

p_{1}(t) = 1, p_{2}(t)=t^{2} and p_{3}(t) = 3+3\cdot t +t^{2}

\alpha_{1}\cdot p_{1}(t) + \alpha_{2}\cdot p_{2}(t) + \alpha_{3}\cdot p_{3}(t) = 0

\alpha_{1}\cdot 1 + \alpha_{2}\cdot t^{2}+ \alpha_{3}\cdot (3+3\cdot t+t^{2}) = 0

(\alpha_{1}+3\cdot \alpha_{3})\cdot 1+(\alpha_{2}+\alpha_{3})\cdot t^{2}+3\cdot \alpha_{3}\cdot t = 0

The following system of linear equations is obtained:

\alpha_{1}+3\cdot \alpha_{3} = 0

\alpha_{2} + \alpha_{3} = 0

3\cdot \alpha_{3} = 0

Whose solution is \alpha_{1} = \alpha_{2} = \alpha_{3} = 0, which means that the set of vectors is linearly independent.

The set of vectors A and C are linearly independent.

You might be interested in
2(11x-6)-18 is equal to 22x-30
rewona [7]

Answer: Yes they are equal

Step-by-step explanation: First you have to solve 2(11x-6)-18:

2(11x-6)-18

22x - 12 - 18

22x - 30

They equal!

7 0
3 years ago
Read 2 more answers
Which ordered pairs in the form (x, y)(x, y) are solutions to the equation
kap26 [50]
The answer is (-1,-7) because you are pluging it in -1 is x and -7 is y. so 7×-1=-7 and 5×-7 = -35 so -7 minus -35 =28
6 0
3 years ago
Can someone help me please and thank you very much
GaryK [48]

Answer:

7-(-3)= 10

7-3=4

59-(-1)= 60

-17-(6)= -11

Step-by-step explanation:

Hope this helps! :3

7 0
3 years ago
GameStop was selling Ps5’s. They sold 100. They sold each ps5 for 500. But they got ps5’s on discount $30 off from each GameStop
Cerrena [4.2K]
They made 3000$ in profit 500x100=50k

470x100=47k
47,000-50000=3000
3 0
3 years ago
A man has 1020304 eggs some of the eggs were spoiled he divided remaining eggs equally among 348 people how many eggs were spoil
beks73 [17]

Answer:

316 eggs were spoiled.

Step-by-step explanation:

1020304 / 348 = 2931 remainder 316.

6 0
3 years ago
Other questions:
  • Couldn't find science but this is a science question.
    12·1 answer
  • What times itself equals 30
    12·2 answers
  • The main goal of personal financial planning is: saving and investing for future needs. reducing a person's tax liability. achie
    10·1 answer
  • Please help. what is the quotient of 4m^12/x-1÷x^2/8m^3 assume x ≠ 0 and m≠0
    10·1 answer
  • Can someone help me find the slope shown on this line please
    6·1 answer
  • Graph the equations to solve the system. y = x + 4 y = -x - 6 choose the correct answer a.) solutions: all numbers on the line
    10·1 answer
  • Kim ordered a few appetizers to share with friends. The total of her meal was
    15·1 answer
  • What are the zeros of the quadratic function f(x) = 6x2 – 24x + 1?
    13·1 answer
  • Which list shows all the factors of 36
    11·1 answer
  • Jax is proving that the base angles of an isosceles triangle are congruent. He joins a line from
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!