# Linearly Dependent

• May 19th 2011, 01:56 PM
barhin
Linearly Dependent
Dear All,
I am trying to solve these following problem i pick from a book i am confuse as to how to approach the question i will appreciate your help or solution

Thank you

Question
Which of the following sets of vectors in P
2 are linearly dependent?

1. 2-x+4x2, 3+6x +2x2, 2+10x - 4x2
2. 3+x+x2, 2-x+5x2, 4-3x2
3. 6-x2, 1+x+4x2
4. 1+3x+3x2,x+4x2,5+6x+3x2,7+2x-x2

NB:X2 the 2 infront of the x's are superscripts
• May 19th 2011, 02:30 PM
TheEmptySet
Quote:

Originally Posted by barhin
Dear All,
I am trying to solve these following problem i pick from a book i am confuse as to how to approach the question i will appreciate your help or solution

Thank you

Question
Which of the following sets of vectors in P
2 are linearly dependent?

1. 2-x+4x2, 3+6x +2x2, 2+10x - 4x2
2. 3+x+x2, 2-x+5x2, 4-3x2
3. 6-x2, 1+x+4x2
4. 1+3x+3x2,x+4x2,5+6x+3x2,7+2x-x2

NB:X2 the 2 infront of the x's are superscripts

$\mathbb{P}_2$

is the vector space of all polynomials of degree 2 or less.

This is isomorphic to

$\mathbb{P}_2 \cong \mathbb{R}$

So

$2-x+4x^2 \iff 2\mathbf{i}-1\mathbf{j}+4\mathbf{k}$

$3+6x+2x^2 \iff 3\mathbf{i}+6\mathbf{j}+2\mathbf{k}$

$2+10x-4x^2 \iff 2\mathbf{i}+10\mathbf{j}+4\mathbf{k}$

Can you finish from here
• May 19th 2011, 03:43 PM
HallsofIvy
Since you are attempting to do a problem involving "dependent" and "independent" vectors you surely must have seen the definition of "dependent" so just check whether those are satisfied.

Do there exist three non-zero numbers a, b, and c, such that a(2-x+4x^2)+ b(3+6x +2x^2)+ c(2+10x - 4x^2)= 0. One way to do that to collect the coefficients of the same powers and set the coefficients equal to 0 to get three equations in a, b, and c. Another way would be to set x equal to any three numbers you like to get to get three equations.
• May 21st 2011, 06:09 AM
barhin
Dear hallsoflvy,
Thanks for suggestions but will appreciate if you could throw more light on the results for me or if possible solution to the question.
• May 21st 2011, 06:15 AM
alexmahone
For 1) and 2), check if the determinant of the 9 components is zero.

For 3), if one vector is not a multiple of the other, they are linearly independent.

For 4), since you have 4 vectors in a 3-dimensional space, they must be linearly dependent.