Thread: 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 P2 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

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 P2 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

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

This is isomorphic to



So








Can you finish from here

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.

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.

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.