Find Linear Combinations for {1, x, x^2, x^3}

In summary, to find the linear combinations that will give you 1, x, x^2, x^3, you can set up a matrix and augment it with the desired answers. Row reduction can then be used to solve for the appropriate constants. Writing out x(x-1) as a simpler expression may also make the process easier.
  • #1
FrostScYthe
80
0
{1, x, x(x-1), x(x-1)(x-2)} you want to find the linear combinations that will give you 1, x, x^2, x^3

a + a(x) + a(x(x-1)) + a(x(x-1)(x-2)) = 1
a + a(x) + a(x(x-1)) + a(x(x-1)(x-2)) = x
a + a(x) + a(x(x-1)) + a(x(x-1)(x-2)) = x^2
a + a(x) + a(x(x-1)) + a(x(x-1)(x-2)) = x^3

I don't know what to do from there
 
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  • #2
I think you might not want to call the constants all 'a'.

1=1
x=x
as for the others trying writing out x(x-1) as something easier to work with
 
  • #3
If you set up an appropriate matrix and augmenting it with the answers you want, row reduction should give you your answers.
 

FAQ: Find Linear Combinations for {1, x, x^2, x^3}

What is a linear combination?

A linear combination is a mathematical operation where two or more terms are multiplied by constants and added together. In this context, we are looking for a combination of the terms 1, x, x^2, and x^3 that can be used to represent any polynomial function of x.

Why is finding linear combinations useful?

Finding linear combinations allows us to express more complex functions in terms of simpler ones, which can make them easier to work with and analyze. It also helps us identify patterns and relationships between different terms in a function.

How do I find linear combinations for {1, x, x^2, x^3}?

To find linear combinations for {1, x, x^2, x^3}, we can start by writing out the coefficients for each term: 1, a, b, and c. Then, we can set up a system of equations by equating it to a general polynomial function of x, such as ax^3 + bx^2 + cx + d. By solving this system of equations, we can find the coefficients that represent the linear combinations for {1, x, x^2, x^3}.

What is the basis of a linear combination?

The basis of a linear combination is the set of terms that are used to form the combination. In this case, the basis is {1, x, x^2, x^3}.

Can I use other terms besides {1, x, x^2, x^3} for finding linear combinations?

Yes, you can use any set of terms to find linear combinations. However, the number of terms in the set will determine the number of coefficients that need to be solved for. The set {1, x, x^2, x^3} is commonly used because it can represent any polynomial function of x.

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