Find The Coordinates Of $p(x) = 3 + 4x + 5x^2$ With Respect To The Basis $B = {1, X, X^2}$.A. (4, 5) B. (0, 4, 5) C. (3, 4, 5)

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Introduction

In the realm of linear algebra, the concept of a basis is crucial in understanding the structure of vector spaces. A basis is a set of vectors that spans the entire space and is linearly independent. When we have a polynomial function, such as $p(x) = 3 + 4x + 5x^2$, we can express it in terms of a basis $B = {1, x, x^2}$. This process involves finding the coordinates of the polynomial with respect to the basis, which is a fundamental problem in mathematics.

What are Coordinates with Respect to a Basis?

The coordinates of a vector with respect to a basis are the coefficients that represent the vector as a linear combination of the basis vectors. In other words, if we have a vector $v$ and a basis $B = {b_1, b_2, ..., b_n}$, then the coordinates of $v$ with respect to $B$ are the scalars $c_1, c_2, ..., c_n$ such that $v = c_1b_1 + c_2b_2 + ... + c_nb_n$.

Finding Coordinates of a Polynomial with Respect to a Basis

To find the coordinates of the polynomial $p(x) = 3 + 4x + 5x^2$ with respect to the basis $B = {1, x, x^2}$, we need to express $p(x)$ as a linear combination of the basis vectors. This can be done by equating the coefficients of the polynomial with the coefficients of the basis vectors.

Let's start by writing the polynomial as a linear combination of the basis vectors:

p(x)=c1(1)+c2(x)+c3(x2)p(x) = c_1(1) + c_2(x) + c_3(x^2)

where $c_1, c_2, c_3$ are the coordinates of $p(x)$ with respect to the basis $B$.

Now, we can equate the coefficients of the polynomial with the coefficients of the basis vectors:

c1=3c_1 = 3

c2=4c_2 = 4

c3=5c_3 = 5

Therefore, the coordinates of the polynomial $p(x) = 3 + 4x + 5x^2$ with respect to the basis $B = {1, x, x^2}$ are $c_1 = 3, c_2 = 4, c_3 = 5$.

Conclusion

In this article, we explored the concept of finding coordinates with respect to a basis and applied it to a polynomial function. We saw that the coordinates of a vector with respect to a basis are the coefficients that represent the vector as a linear combination of the basis vectors. We also demonstrated how to find the coordinates of a polynomial with respect to a basis by equating the coefficients of the polynomial with the coefficients of the basis vectors. The coordinates of the polynomial $p(x) = 3 + 4x + 5x^2$ with respect to the basis $B = {1, x, x^2}$ are $c_1 = 3, c_2 = 4, c_3 = 5$.

Answer

The correct answer is:

C. (3, 4, 5)

Discussion

This problem is a fundamental concept in linear algebra and is used to understand the structure of vector spaces. The coordinates of a vector with respect to a basis are essential in many applications, including computer graphics, data analysis, and machine learning.

In this problem, we used the basis $B = {1, x, x^2}$ to find the coordinates of the polynomial $p(x) = 3 + 4x + 5x^2$. This basis is a standard basis for the vector space of polynomials of degree at most 2.

The coordinates of the polynomial with respect to the basis $B$ are $c_1 = 3, c_2 = 4, c_3 = 5$. This means that the polynomial $p(x)$ can be expressed as a linear combination of the basis vectors:

p(x)=3(1)+4(x)+5(x2)p(x) = 3(1) + 4(x) + 5(x^2)

This representation is useful in many applications, including numerical analysis and approximation theory.

References

  • [1] Linear Algebra and Its Applications, Gilbert Strang
  • [2] Introduction to Linear Algebra, Gilbert Strang
  • [3] Linear Algebra, David C. Lay

Keywords

  • Linear algebra
  • Basis
  • Coordinates
  • Polynomial
  • Vector space
  • Linear combination
    Q&A: Finding Coordinates with Respect to a Basis =============================================

Q: What is a basis in linear algebra?

A: A basis is a set of vectors that spans the entire vector space and is linearly independent. In other words, a basis is a set of vectors that can be used to express any vector in the space as a linear combination of the basis vectors.

Q: What are coordinates with respect to a basis?

A: The coordinates of a vector with respect to a basis are the coefficients that represent the vector as a linear combination of the basis vectors. In other words, if we have a vector $v$ and a basis $B = {b_1, b_2, ..., b_n}$, then the coordinates of $v$ with respect to $B$ are the scalars $c_1, c_2, ..., c_n$ such that $v = c_1b_1 + c_2b_2 + ... + c_nb_n$.

Q: How do I find the coordinates of a vector with respect to a basis?

A: To find the coordinates of a vector with respect to a basis, you need to express the vector as a linear combination of the basis vectors. This can be done by equating the coefficients of the vector with the coefficients of the basis vectors.

Q: What is the difference between a basis and a coordinate system?

A: A basis is a set of vectors that spans the entire vector space and is linearly independent, while a coordinate system is a set of axes that are used to locate points in a space. In other words, a basis is used to express vectors as linear combinations of the basis vectors, while a coordinate system is used to locate points in a space.

Q: Can a basis be used to find the coordinates of a vector in a higher-dimensional space?

A: Yes, a basis can be used to find the coordinates of a vector in a higher-dimensional space. In fact, a basis is a fundamental tool in linear algebra for expressing vectors in higher-dimensional spaces.

Q: How do I choose a basis for a vector space?

A: Choosing a basis for a vector space involves selecting a set of vectors that spans the entire space and is linearly independent. This can be done by selecting a set of vectors that are linearly independent and span the space.

Q: What are some common bases used in linear algebra?

A: Some common bases used in linear algebra include the standard basis, the monomial basis, and the trigonometric basis. The standard basis is a set of vectors that are used to express vectors in a space, while the monomial basis and the trigonometric basis are used to express vectors in terms of monomials and trigonometric functions, respectively.

Q: Can a basis be used to find the coordinates of a polynomial?

A: Yes, a basis can be used to find the coordinates of a polynomial. In fact, a basis is a fundamental tool in linear algebra for expressing polynomials in terms of the basis vectors.

Q: How do I find the coordinates of a polynomial with respect to a basis?

A: To find the coordinates of a polynomial with respect to a basis, you need to express the polynomial as a linear combination of the basis vectors. This can be done by equating the coefficients of the polynomial with the coefficients of the basis vectors.

Q: What are some common bases used to express polynomials?

A: Some common bases used to express polynomials include the monomial basis, the power basis, and the Chebyshev basis. The monomial basis is a set of vectors that are used to express polynomials in terms of monomials, while the power basis and the Chebyshev basis are used to express polynomials in terms of powers and Chebyshev polynomials, respectively.

Q: Can a basis be used to find the coordinates of a vector in a non-Euclidean space?

A: Yes, a basis can be used to find the coordinates of a vector in a non-Euclidean space. In fact, a basis is a fundamental tool in linear algebra for expressing vectors in non-Euclidean spaces.

Q: How do I choose a basis for a non-Euclidean space?

A: Choosing a basis for a non-Euclidean space involves selecting a set of vectors that spans the entire space and is linearly independent. This can be done by selecting a set of vectors that are linearly independent and span the space.

Q: What are some common bases used in non-Euclidean spaces?

A: Some common bases used in non-Euclidean spaces include the orthonormal basis, the spherical basis, and the cylindrical basis. The orthonormal basis is a set of vectors that are used to express vectors in a non-Euclidean space, while the spherical basis and the cylindrical basis are used to express vectors in terms of spherical and cylindrical coordinates, respectively.