# induction proof vectors

• Mar 6th 2007, 12:22 PM
faure72
induction proof vectors
i have to do a lin. algebra induction proof, and i'm not sure how to go about it. here it is:

let x1,....xn+1 (that means x sub 1 through x sub n+1...sorry, i dont know how to do subscripts on this thing) be vectors in the set of real numbers.

(a) show that there exist real numbers a1,....an+1, not all zero, such that the linear combination a1x1 +....+an+1 xn+1 = 0.

(b) using part a, show that

xi (again, meaning x sub i) = b1x1 +....+bi-1 xi-1 +...+bn+1 xn+1, for some i, 1 is less than or equal to i, which is less than or equal to n+1, and some b1,...bi-1, bi+1,...bn+1 is in the set of real numbers.
• Mar 6th 2007, 01:25 PM
ThePerfectHacker
Quote:

Originally Posted by faure72
i have to do a lin. algebra induction proof, and i'm not sure how to go about it. here it is:

let x1,....xn+1 (that means x sub 1 through x sub n+1...sorry, i dont know how to do subscripts on this thing) be vectors in the set of real numbers.

(a) show that there exist real numbers a1,....an+1, not all zero, such that the linear combination a1x1 +....+an+1 xn+1 = 0.

(b) using part a, show that

xi (again, meaning x sub i) = b1x1 +....+bi-1 xi-1 +...+bn+1 xn+1, for some i, 1 is less than or equal to i, which is less than or equal to n+1, and some b1,...bi-1, bi+1,...bn+1 is in the set of real numbers.

Are your trying to show that if a_1,a_2,...,a_n spam V then a_1,a_2,...,a_n,a_{n+1} are linearly dependent for any a_{n+1} element in V.
• Mar 6th 2007, 01:31 PM
faure72
yes, i believe that is what i'm trying to show.
• Mar 6th 2007, 01:49 PM
ThePerfectHacker
Quote:

Originally Posted by faure72
yes, i believe that is what i'm trying to show.

You can express,
a_{n+1}=b_1*a_1+...+b_n*a_n

Because {a_1,...,a_n} for a spam set.

That means,
k_1*a_1+...+k_n*a_n+k_{n+1}*a_{m+1}=0
Can be replaced by,
k_1*a_1+...+k_n*a_n+k_{n+1}*(b_1*a_1+...+b_n*a_n)= 0
And show that it can have a non-trivial representation.