## atjari 3 years ago Pls help. see the attachment for the question.

1. atjari

2. atjari

@JamesJ can u help me pls?

3. JamesJ

Please is such an underused work on this site.

4. JamesJ

Let u=i+j+2k, v=3i+j-k and w=βi+(β-1)j-k where β is a real number. Find β such that the space spanned by u and v is the same as the space spanned by u and w.

5. JamesJ

*word

6. atjari

I am sorry I didn't get you.

7. JamesJ

I am saying thank you for asking with "please", vs. just putting out a demand that I or someone else help you. It's slightly ironic. Don't worry about it. Down to business. If u and v span the same space as u and w, then they will have the same row reduction.

8. JamesJ

I.e., the row reduction of 1 1 2 β β-1 -1 will be the same as the row reduction of 3 1 -1 β β-1 -1 Use that fact to find a value of β for which that is true.

9. atjari

If you dnt mind can u tel me vat is row deduction?

10. JamesJ

Ok, you don't that. Another strategy then. If the span of the two pairs of vectors is the same, then any vector in one span can be written as the vector in the other. Hence for any vector in the span of u and w, i.e., x = au + bw where a and b are arbitrary constants. x is also a member of the span of v and w; i.e., there exist constants c and d such that x = cv + dw

11. JamesJ

Now equate those two expressions for the vector x and find what what implies for beta.

12. JamesJ

You will end up with three equations, one for each component of x. Remember that a and b are arbitrary. This constrains your choice of beta.

13. atjari

Nw I understand it . Thanx a lot. was the way I askd u for the help wrong? If so I am extremely sorry.

14. JamesJ

15. atjari

Thanx a lot again.

16. atjari

hw do arbitary constants limit the choice of beta

17. atjari

|dw:1331146251985:dw| After this I am stuck. Can you please tell me how to proceed after this?

18. JamesJ

Think of this now as a system of equations in three variables: c, d and beta. a and b are just parameters. Now ask yourself the question: what is the solution to this system of equations in those three variables?

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