# A subspace spanned by polynomials 1 and x

• beramodk
In summary, the conversation discusses finding an orthogonal projection of the polynomial p(x) = 1+x^2 to the linear subspace W, which is spanned by the polynomials 1 and x. The basis for the orthogonal projection is found to be {1,x} and the inner product is defined as <p(x),q(x)> = integral (p(x)*q(x), x, 0, 1). The process for solving this problem involves setting up two linear equations in the unknowns a and b.
beramodk
1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp)

My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula.

Would it simply be [1, x] or [1 ; x]?

Thanks!

{1,x} is certainly a basis for W. But to define orthogonal you need to say what the inner product is.

Last edited:
Remember, the dot product (or inner product) should be equal to 0 if they are orthogonal.

Dick said:
{1,x} is certainly a basis for W. But to define orthogonal you need to say what the inner product is.

If the inner product is given by: <p(x),q(x)> = integral (p(x)*q(x), x, 0, 1), how would i go about solving this problem?

(This is not my homework by the way, I'm studying for my final exam which is tomorrow, but I just can't figure out this problem.)

Thank you.

If a*1+b*x is a point in W that is the orthogonal projection, you want that <(1+x^2)-(a*1+b*x),1>=0 and <(1+x^2)-(a*1+b*x),x>=0. That would say that the difference between (1+x^2) and (a*1+b*x) is perpendicular to all of the vectors in W which is the span of {1,x}, wouldn't it? It's just two linear equations in the unknowns a and b.

## 1. What is a subspace spanned by polynomials 1 and x?

A subspace spanned by polynomials 1 and x is a mathematical concept in linear algebra where the set of all possible linear combinations of the polynomials 1 and x form a vector space. This means that any polynomial that can be expressed as a combination of 1 and x, with coefficients from the underlying field, is considered to be a part of this subspace.

## 2. How is a subspace spanned by polynomials 1 and x different from a regular vector space?

A subspace spanned by polynomials 1 and x is a specific type of vector space. While a regular vector space can have any set of linearly independent vectors as its basis, a subspace spanned by polynomials 1 and x has a specific basis of only the polynomials 1 and x.

## 3. What is the dimension of a subspace spanned by polynomials 1 and x?

The dimension of a subspace spanned by polynomials 1 and x is 2. This is because the two polynomials, 1 and x, form a basis for this subspace. Any polynomial of the form a + bx, where a and b are coefficients, can be expressed as a linear combination of these two polynomials.

## 4. Can a subspace spanned by polynomials 1 and x contain polynomials of higher degree?

Yes, a subspace spanned by polynomials 1 and x can contain polynomials of higher degree. This is because any polynomial of higher degree can be reduced to a linear combination of 1 and x. For example, x^2 can be expressed as 0 + x^2.

## 5. What are some real-world applications of a subspace spanned by polynomials 1 and x?

A subspace spanned by polynomials 1 and x has various applications in fields such as physics, engineering, and computer science. In physics, it can be used to model motion or forces acting on a system. In engineering, it can be used to solve problems in structural analysis or fluid mechanics. In computer science, it can be used in algorithms for pattern recognition or data compression.

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