bob258173498 said:Show that for any three distinct values a,b,c, the vectors (1,a,a2), (1,b,b2), (1,c,c2) are linearly independent.
bob258173498 said:Homework Statement
Find a basis for (1, a, a^2) (1, b, b^2) (1, c, c^2)
Homework Equations
The Attempt at a Solution
M(1, a, a^2) + N(1, b, b^2) + K(1, c, c^2) = (0, 0, 0)
M + N + K = 0
Ma + Nb + Kc = 0
Ma^2 + Nb^2 + Kc^2 = 0
This is as far as I got. I tried monkeying around with these 3 equations to no avail.
Yes, it will work in this case.bob258173498 said:Can someone please just show me how to solve it. I don't know how to solve it when the variables are general and not actual numbers. Normally I just get rid of one variable by mult/subtracting equations... but it won't work in this case. Please help!
A basis for a vector space is a set of vectors that are linearly independent and span the entire vector space. In other words, any vector in the space can be written as a linear combination of the basis vectors.
To find a basis for a vector space, you can use the process of elimination. Start by writing the given vectors as columns in a matrix and then use elementary row operations to reduce the matrix to row-echelon form. The non-zero rows in the reduced matrix will form the basis for the vector space.
No, not all sets of vectors can be a basis for a vector space. The set of vectors must be linearly independent and span the entire vector space in order to be considered a basis.
The number of vectors needed to form a basis for a vector space is equal to the dimension of the vector space. In this case, since the vectors are in three-dimensional space, we need three linearly independent vectors to form a basis.
Yes, the basis for a vector space can be unique. However, there can also be multiple sets of vectors that form a basis for a vector space. It is possible for a vector space to have more than one basis.