Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

RolyPoly

  • 2 years ago

Concept-checking questions Consider vectors \(\vec{u}\), \(\vec{v}\) and \(\vec{w}\), if \(\vec{w}\) is a linear combination of \(\vec{u}\) and \(\vec{v}\), does that mean that i) \(\vec{w}\) is linearly dependent on \(\vec{u}\) and \(\vec{v}\) ? ii) sp(\(\vec{u}\), \(\vec{v}\), \(\vec{w}\)) = sp(\(\vec{u}\), \(\vec{v}\)) ?

  • This Question is Closed
  1. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    (i) follows straight from the definition of linear dependence :-) For (ii), recognize that the linear span of a set of vectors is the set of all its finite linear combinations; if \(\vec{u},\vec{v},\vec{w}\) are linearly dependent, i.e. \(\vec{w}\) is a linear combination of \(\vec{u},\vec{v}\), then it should be immediately apparent that the spanning sets are identical.

  2. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    If I can show \(\vec{u}=(u_1, u_2, u_3)\), \(\vec{v}=(v_1, v_2, v_3)\), and \(\vec{w}=(w_1, w_2, w_3)\) span \(\Re^3\), can I immediately draw a conclusion that the vectors are linearly independent?

  3. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I believe so, yes...

  4. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes yes yes.

  5. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    And so, if they do not span \(\Re^3\), they are linearly dependent, right?

  6. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah three vectors that aren't all linearly independent cannot span \[\mathbb R^3\]

  7. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Could "yeah three vectors that aren't all linearly independent cannot span \(\Re^3\)" be the justification of the answer?

  8. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i wouldn't word it like that

  9. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Since it cannot span \(\Re^3\), that means one of the vectors is a linear combination of the other vectors, therefore, the vectors are linearly dependent. ^ Could that be one?

  10. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    thats is better yes

  11. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That's pretty nice :) Thanks! May I ask one more question about linear dependence here?

  12. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    sure

  13. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Checking the linear dependence of vectors, say, \(\vec{u} =(u_1,u_2,u_3), \vec{v} =(v_1,v_2,v_3),\) and \(\vec{w} =(w_1,w_2,w_3)\) Method I) \[\left[\begin{matrix}u_1 & v_1 & | & w_1 \\ u_2 & v_2 & | & w_2 \\u_3 & v_3 & | &w_3\end{matrix}\right]\] Solve it, if it is a) consistent (that is it has solution), then it is linearly dependent b) inconsistent (that is it has no solution), the it is linearly independent Method II) Find the determinant of \[\left[\begin{matrix}u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\u_3 & v_3 & w_3\end{matrix}\right]\] If the determinant is a) equal to 0, it is linearly dependent b) not equal to 0, it is linearly independent Is that correct? Is there any other way to check the linear dependence?

  14. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    for the first method if (0,0,0) is the only solution the the vectors are independent the second method is right, and im sure there are other methods but i can think of them

  15. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I think for the first method, it's actually testing if \(\vec{w}\) is a linear combination of \(\vec{u}\) and \(\vec{v}\) Please leave a comment here if you can think of any of them. That would be really helpful! Thank you for your help!! :)

  16. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If \(\vec{w}\) can be written as a linear combination of \(\vec{u}, \vec{w}\) then they are dependent.

  17. oldrin.bataku
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \(\vec{v}\) **

  18. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    And that's why "if it is a) consistent (that is it has solution), then it is linearly dependent b) inconsistent (that is it has no solution), the it is linearly independent" :|

  19. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    if you find the angle between vectors is π/2 the vectors will be independent

  20. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    How to find the angle between three vectors??

  21. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[\theta=\arccos\left(\frac{\vec u \cdot \vec v}{||\vec u||~~||\vec v||}\right)\]

  22. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    For multiple vectors you would have to find the angle between each possible pair of vectors

  23. RolyPoly
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    For the case of two vectors, I still can do it. But not for three or more vectors..

  24. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.