Question

Find three functions u1,u2,u3 which are linearly dependent and such that u1 can't be expressed as a a linear combination of u2 and u3

Answer 1

Can't I just put that for u2 and u3... let u2 = [1,2,3] and u3 = 2[1,2,3]
because u3 has a multiple of 2 and it would be u3 = [2.4.6]

Answer 2

u1 can't be related at all. so let u1 = [1.5.6]

Answer 3

or if there is a vector such that [3.4.5]
u2 + u3
u2 = [ 1.2.3] and u3=[2,2,2]
u2+u3=[3,4,5]

Answer 4

@KingGeorge

Answer 5

You have the right idea. u2 and u3 should be linearly dependent, and u1 needs to be linearly independent from u2 and u3, and the first example you had would work as a vector.

Answer 6

Although in this case it asks for functions, so I might consider using something like \(x\) and \(x^2\) instead of regular old vectors.

Answer 7

danggggg but these questions are so similar to linear algebra. why can't i just use linear algebra definitions since that Worksikins thing yeah spelling is using a matrix, so the first thing that comes to my mind. oh linear algebra something I took last semester.

Answer 8

You actually can consider functions as different vectors. For example, \(x^n\) and \(x^k\) are linearly independent for \(n\neq k\). Also, trig functions can be made to be linearly independent as well.

Answer 9

In fact, one of the more common function vector spaces people use from time to time, is the set of real valued continuous functions on the interval \([a,b]\). This vector space has infinite dimension, and has various different bases you can choose from.