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atjari Group Title

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

  • 2 years ago
  • 2 years ago

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  1. JamesJ Group Title
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    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 years ago
  2. atjari Group Title
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    since alpha is unknown v cn't v shw that it has non zero determinant. isnt t?

    • 2 years ago
  3. atjari Group Title
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    for 2 values of alpha the matrix becoes zero. then hw do v shw?

    • 2 years ago
  4. atjari Group Title
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    James pls help.

    • 2 years ago
  5. JamesJ Group Title
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    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.

    • 2 years ago
  6. atjari Group Title
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    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."

    • 2 years ago
  7. JamesJ Group Title
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    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 \[ p = \frac{53 - 2\sqrt{115}}{2(\sqrt{115}-22)}, q = \frac{\sqrt{115} -4}{2(\sqrt{115}-22}\]

    • 2 years ago
  8. atjari Group Title
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    how do u get this value for alpha? I get alpha as +/-sqrt-7+2

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

    • 2 years ago
  10. atjari Group Title
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    then hw do v shw that it is nt in the space?

    • 2 years ago
  11. JamesJ Group Title
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    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.

    • 2 years ago
  12. atjari Group Title
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    extremly sorry 4 troubling u a lt.

    • 2 years ago
  13. atjari Group Title
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    Thanx a lt 4 ur great help.

    • 2 years ago
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