Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

atjari

  • 2 years ago

Let vectora=-4i+3j-alphak, vector b=2i+alphaj+k and vectorc=5i-j+alphak, where alpha is a real number. Show that the vector c is not in the space spanned by vectors a and b. Pls help

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

    c is not in the span of a and b, if and only if the three vectors a,b,c are linearly independent. That means that the system of coefficients of the three vectors has non-zero determinant. So one way to show the result of the question is to show that the matrix -4 3 a 2 a 1 5 -1 a has non-zero determinant.

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

    since alpha is unknown v cn't v shw that it has non zero determinant. isnt t?

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

    for 2 values of alpha the matrix becoes zero. then hw do v shw?

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

    James pls help.

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

    Yes, you're exactly right. For two values of a, the determinant IS zero and hence the vector c DOES lie in the span of the vectors a and b. So the question as written is not exactly correct.

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

    if I solve c=pa+qb i get alpha as an imaginary number. so cn v say that "since alpha is given as real number bt v get here alpha as an imaginary number. so c is nt in the spce spand by a nd b."

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

    It's a brutal calculation, but if \[ \alpha = \frac{1}{9} (-4 \pm \sqrt{115} ) \] then there are real numbers p and q such that c = pa + qb. For example, if alpha = (1/9) ( -4 + sqrt(115)), then \[ p = \frac{53 - 2\sqrt{115}}{2(\sqrt{115}-22)}, q = \frac{\sqrt{115} -4}{2(\sqrt{115}-22}\]

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

    how do u get this value for alpha? I get alpha as +/-sqrt-7+2

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

    the determinant of the matrix written above is \[ -9\alpha^2 - 8\alpha + 11 \] That expression is zero, when \( \alpha \) is \[ \alpha = \frac{-4 \pm \sqrt{115}}{9} \]

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

    then hw do v shw that it is nt in the space?

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

    As I noted above, the question as written is wrong. For two values of alpha, the vector c DOES lie in span of the vectors a and b.

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

    extremly sorry 4 troubling u a lt.

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

    Thanx a lt 4 ur great help.

  14. 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.