From a given basis, express a polynomial

[Randell Julius]

Homework Statement

Express a polynomial in terms of the basis vectors.
{x² + x, x + 1, 2}

Homework Equations

3. The Attempt at a Solution [/B]
I think the answer is:
(x²+x)^2 + (x + 1) + 2 = 0
simplified to become:
x⁴ + 2x³ + x² + x + 3 = 0

 

[member 587159]

What polynomial do you have to express in that basis? Is this a linear algebra question? In a vector space, there is no such thing as a product of two polynomials, so if this is a linear algebra question this doesn't make sense.

 

[Mark44]

Homework Statement

Express a polynomial in terms of the basis vectors.
{x² + x, x + 1, 2}

Homework Equations

3. The Attempt at a Solution [/B]
I think the answer is:
(x²+x)^2 + (x + 1) + 2 = 0
simplified to become:
x⁴ + 2x³ + x² + x + 3 = 0

This answer makes no sense for a couple of reasons.
1. Your polynomial will necessarily be of degree at most 2.
2. The answer should be a polynomial, not an equation in which the polynomial equals zero.

Whatever your polynomial is (which you didn't include), it should be written as a linear combination of your basis functions. By linear combination, I mean a sum of constant multiples of those basis functions $x^2 + x, x + 1,$ and $2$.

Please include the full statement of the problem you're working on.

 

[Randell Julius]

This answer makes no sense for a couple of reasons.
1. Your polynomial will necessarily be of degree at most 2.
2. The answer should be a polynomial, not an equation in which the polynomial equals zero.

Whatever your polynomial is (which you didn't include), it should be written as a linear combination of your basis functions. By linear combination, I mean a sum of constant multiples of those basis functions $x^2 + x, x + 1,$ and $2$.

Please include the full statement of the problem you're working on.

This is what I initially thought it would be, but this is all that my professor gave me. I will ask him.

Thank you.

 

[Mark44]

Perhaps he means "Express a quadratic polynomial $ax^2 + bx + c$ in terms of the basis functions $x^2 + x, x + 1,$ and $2$."

 

[Physics345]

Express a polynomial in terms of the basis vectors.
{x² + x, x + 1, 2}

What do vectors have to do with this at all?
{x2 + x, x + 1, 2}

This is not in vector format if anything it's what mark said "a quadratic polynomial" (which means the function has a degree of two)
I am absolutely positive a professor would not give out a question like that.

 

[Mark44]

What do vectors have to do with this at all?

The functions $x^2 + x, x + 1$, and $2$ are a basis for $P_2(x)$, the space of functions of degree 2 or less. In other words, the set of all polynomials of the form $ax^2 + bx + c$, with a, b, and c being real numbers. A function space is almost identical to a vector space; in this case $P_2(x)$ is isomorphic (same "shape") as $\mathbb R^3$ -- each polynomial is paired to a specific vector in $\mathbb R^3$ and vice versa. The operation of addition of polynomials corresponds to addition of vectors, and scalar multiplication of a polynomial corresponds to scalar multiplication of vectors.

 

[Ray Vickson]

What do vectors have to do with this at all?
{x2 + x, x + 1, 2}

This is not in vector format if anything it's what mark said "a quadratic polynomial" (which means the function has a degree of two)
I am absolutely positive a professor would not give out a question like that.

The correspondence
$$a + b x +c x^2 \leftrightarrow (a,b,c)$$
turns the space of second-degree polynomials into a three-dimensional vector space. The sum of two polynomials turns into the sum of two vectors; the product of a polynomial and a number turns into the product of a vector and a number. The space of polynomials IS a vector space.