Advanced Math/Basis

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QUESTION: I a confused by this problem. For part a) i know how to do the dot product but i don't know what to do for part b) . Any help would be appreciated.

Consider the vectors vector u = < 1,-1, 3 >, vetor v = < 1, k, k >, vector w =< k, 1, k^2 >

(a) Compute vector u dot product((vector v)cross product(vector w)).
(b) For which values of k is {< 1,-1, 3 >,< 1, k, k >,< k, 1, k^2 >} a basis for R^3?

ANSWER: For part b, the vectors must be linearly independent to be a basis, so that the determinant of matrix made up of the vectors must be non-zero.

---------- FOLLOW-UP ----------

QUESTION: For part a) i got the cross product to be (k^3-k)+(1-k^2)
Then dot producting it with u resulted in (k^2-1)(k-3)
Is this right?

And for part b) i got the determinant to have a solution to be (k^2-1)(k-3). This resulted in k being equal to -1,1,3. It must be equal to any values but these three to form the basis. I was just wondering if there was another way to solve the problem without using the determinant method because my teacher said she will not accept that method since she has not yet taught it. Thanks for the help.

ANSWER: This begs the question, what method did your teacher teach already? Tell me and I'll take it from there.

---------- FOLLOW-UP ----------

QUESTION: She taught the gaussian elimination. That is what she uses to solve the matrices

Answer
Nick, I re-responded to your previous question, so please look for that. It involves the solution not using determinants but also avoiding gaussian elimination. Hope this works for you. Send me a follow-up if I've managed to totally confuse you.

Randy

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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