Question

Show that $$ \left\\{a+b x+c x^{2}, a_{1}+b_{1} x+c_{1} x^{2}, a_{2}+b_{2} x+c_{2} x^{2}\right\\} $$ is a basis of \(\mathbf{P}_{2}\) if and only if \(\left\\{(a, b, c),\left(a_{1}, b_{1}, c_{1}\right),\left(a_{2}, b_{2}, c_{2}\right)\right\\}\) is a basis of \(\mathbb{R}^{3}\).

Short answer

The polynomial set is a basis of \( \mathbf{P}_{2} \) if and only if the vector set is a basis of \( \mathbb{R}^{3} \).

Step-by-step solution

Understanding the Polynomial Space

The space \( \mathbf{P}_{2} \) represents all polynomials of degree at most 2. Any polynomial in this space can be written as \( a + bx + cx^2 \), where \( a, b, c \) are coefficients from the set of real numbers, \( \mathbb{R} \).

Expressing Polynomials as Vectors

Each polynomial \( a + bx + cx^2 \) can be associated with a vector \( (a, b, c) \) in \( \mathbb{R}^3 \). This is because the coefficients of the polynomial can be viewed as components of a vector in three-dimensional space.

Basis of \( \mathbf{P}_{2} \)

A set of polynomials forms a basis of \( \mathbf{P}_{2} \) if they are linearly independent and span \( \mathbf{P}_{2} \). This means any polynomial in \( \mathbf{P}_{2} \) can be expressed as a linear combination of these basis polynomials.

Basis of \( \mathbb{R}^{3} \)

Similarly, a set of vectors \( \{(a, b, c), (a_{1}, b_{1}, c_{1}), (a_{2}, b_{2}, c_{2})\} \) forms a basis of \( \mathbb{R}^3 \) if the vectors are linearly independent and span the space \( \mathbb{R}^3 \).

Showing Equivalence

To show that our polynomial set forms a basis for \( \mathbf{P}_{2} \) if and only if the vector set forms a basis for \( \mathbb{R}^{3} \), we use their transformations: \( a + bx + cx^2 \) to \( (a, b, c) \). Both requirements of being a basis (linear independence and spanning) must hold. This equivalence exploits the one-to-one correspondence between polynomial coefficients and vectors.

Linear Independence

For linear independence, polynomials \( a+b x+c x^{2}, a_{1}+b_{1} x+c_{1} x^{2}, a_{2}+b_{2} x+c_{2} x^{2} \) are independent if \( k_1(a, b, c) + k_2(a_1, b_1, c_1) + k_3(a_2, b_2, c_2) = (0, 0, 0) \) implies \( k_1 = k_2 = k_3 = 0 \). This mirrors linear independence of vectors in \( \mathbb{R}^3 \).

Spanning the Space

Spanning \( \mathbf{P}_{2} \) is equivalent to covering all polynomials \( a + bx + cx^2 \). For vectors, \( (a, b, c) \) form every point in \( \mathbb{R}^3 \) if they span the space. Thus, if our polynomials are expressed as linear combinations covering anything in \( \mathbf{P}_{2} \), their vector equivalents span \( \mathbb{R}^3 \).

Conclusion

Since transformation preserves both linear independence and spanning properties, \( \{a + bx + cx^2, a_1 + b_1x + c_1x^2, a_2 + b_2x + c_2x^2\} \) is a basis for \( \mathbf{P}_{2} \) if and only if \( \{(a, b, c), (a_1, b_1, c_1), (a_2, b_2, c_2)\} \) is a basis for \( \mathbb{R}^{3} \).

Key concepts

Polynomial Basis

Polynomials can be represented as a set of basic building blocks known as a polynomial basis. A basis in the polynomial space \( \mathbf{P}_{2} \) consists of polynomials that are linearly independent and span the entire space. Every polynomial in \( \mathbf{P}_{2} \), which is the space of polynomials of degree at most 2, can be expressed as a combination of these basis polynomials.
For instance, the set \( \{1, x, x^2\} \) serves as a standard basis for \( \mathbf{P}_{2} \) since any quadratic polynomial \( ax^2 + bx + c \) can be written as a sum of multiples of these elements. When constructing or understanding any polynomial space, identifying a suitable basis is essential as it simplifies many operations, such as addition, subtraction, and differentiation of polynomials within that space. This concept is analogous to how a set of unit vectors forms a basis in vector spaces of Euclidean geometry.

Vector Spaces

Vector spaces provide a framework for understanding many different mathematical structures. A vector space is a collection of vectors where you can add any two vectors together and scale them by numbers, known as scalars.
The space \( \mathbb{R}^{3} \), for example, is a three-dimensional vector space where each vector has three real-number components. Elementary operations such as vector addition and scalar multiplication always result in another vector within the same space.

• Each polynomial \( a + bx + cx^2 \) can be mapped to a vector \( (a, b, c) \) in \( \mathbb{R}^{3} \), linking concepts of polynomials and vectors.

Understanding vector spaces is crucial as they lay the groundwork for exploring linear transformations, eigenvalues and eigenvectors, and more advanced algebraic concepts.

Linear Independence

A critical property of a basis is linear independence. Vectors or polynomials are linearly independent if none can be written as a combination of the others. When discussing linear independence in \( \mathbf{P}_{2} \), it's about ensuring no polynomial in the basis can be crafted from the others using coefficient scaling and addition.
Mathematically, the polynomials \( a + bx + cx^2, a_1 + b_1x + c_1x^2, a_2 + b_2x + c_2x^2 \) are linearly independent if the only solution to \( k_1(a, b, c) + k_2(a_1, b_1, c_1) + k_3(a_2, b_2, c_2) = (0, 0, 0) \) is when \( k_1=k_2=k_3=0 \).

• This concept ensures that each element of the basis adds something unique, ensuring full coverage of the vector or polynomial space without redundancy.

Without linear independence, the basis would have redundancies, making them inefficient for spanning the space.

Spanning Sets

Spanning refers to a set of vectors or polynomials that can recombine to cover an entire space. For example, a spanning set for \( \mathbf{P}_{2} \) means any polynomial of that form can be written using the set's members.
To say a vector set \( S = \{(a, b, c), (a_1, b_1, c_1), (a_2, b_2, c_2)\} \) spans \( \mathbb{R}^{3} \) implies you can form any vector in the three-dimensional real number space from these three vectors through certain linear combinations.

• Spanning ensures that all possible elements within a vector or polynomial space are achievable through combinations of the basis elements.

Every set of linearly independent vectors in \( \mathbb{R}^{3} \) will span the space, indicating that each vector or polynomial element has a precise purpose in forming the complete structure of the space.