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RolyPoly

  • one year 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(\(\vec{u}\), \(\vec{v}\)) ?

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  1. oldrin.bataku
    • one year ago
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    (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
    • one year ago
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    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
    • one year ago
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    I believe so, yes...

  4. UnkleRhaukus
    • one year ago
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    yes yes yes.

  5. RolyPoly
    • one year ago
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    And so, if they do not span \(\Re^3\), they are linearly dependent, right?

  6. UnkleRhaukus
    • one year ago
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    yeah three vectors that aren't all linearly independent cannot span \[\mathbb R^3\]

  7. RolyPoly
    • one year ago
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    Could "yeah three vectors that aren't all linearly independent cannot span \(\Re^3\)" be the justification of the answer?

  8. UnkleRhaukus
    • one year ago
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    i wouldn't word it like that

  9. RolyPoly
    • one year ago
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    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
    • one year ago
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    thats is better yes

  11. RolyPoly
    • one year ago
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    That's pretty nice :) Thanks! May I ask one more question about linear dependence here?

  12. UnkleRhaukus
    • one year ago
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    sure

  13. RolyPoly
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    If \(\vec{w}\) can be written as a linear combination of \(\vec{u}, \vec{w}\) then they are dependent.

  17. oldrin.bataku
    • one year ago
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    \(\vec{v}\) **

  18. RolyPoly
    • one year ago
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    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
    • one year ago
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    if you find the angle between vectors is π/2 the vectors will be independent

  20. RolyPoly
    • one year ago
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    How to find the angle between three vectors??

  21. UnkleRhaukus
    • one year ago
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    \[\theta=\arccos\left(\frac{\vec u \cdot \vec v}{||\vec u||~~||\vec v||}\right)\]

  22. UnkleRhaukus
    • one year ago
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    For multiple vectors you would have to find the angle between each possible pair of vectors

  23. RolyPoly
    • one year ago
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    For the case of two vectors, I still can do it. But not for three or more vectors..

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