Find a basis of the subspace of http://webwork2.asu.edu/webwork2_fil...8083707491.png consisiting of all vectors of the form

i am not sure how to find this since there are variables and

Thank you for any help :)

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- March 28th 2010, 02:02 PMmybrohshi5[SOLVED] Basis of subspace R
Find a basis of the subspace of http://webwork2.asu.edu/webwork2_fil...8083707491.png consisiting of all vectors of the form

i am not sure how to find this since there are variables and

Thank you for any help :) - March 28th 2010, 03:10 PMBruno J.
Hint :

- March 28th 2010, 03:25 PMmybrohshi5
The basis are just the two you listed because when

is in reduced row echelon form i got

which showed me that

are just the basis of the subspace R.

Does that seem correct?

Thanks for the hint :) - March 28th 2010, 03:43 PMBruno J.
Well it's clear from the hint I gave you that every vector in your subspace can be written as a linear combination of the two vectors above. So the two vectors form a basis if and only if they are linearly independent. It's very easy to check for linear independence when you have only two vectors; two vectors are linearly dependent if and only if one is a multiple of the other. It's clearly not the case here! Your method is also valid (though I'm not sure why you added a third zero column to the matrix you reduced.)

- March 28th 2010, 03:47 PMmybrohshi5
Thank you that makes sense and actually helps me a lot about understanding these.

One quick (and kind of a stupid question) but could you please quickly explain what you mean by "one is a multiple of the other."

I added the extra columns of zero's cause when i plug the matrix into my calculator to get the rref it needed extra columns of zeros in order to calculate it without my calculator saying error (Rofl)

Thanks again - March 28th 2010, 04:55 PMBruno J.
Well, for example, the vectors are linearly dependent because . But the vectors are linearly independent because for all . That's what I mean by one being a multiple of the other.

- March 28th 2010, 08:05 PMmybrohshi5
Perfect thank you so much for the help and explaining that :) i feel silly cause i knew it was easy but i just couldnt remember what a multiple meant haha.