Question

Using the inner product given by \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x\) on \(\mathbf{P}_{2},\) write \(\mathbf{v}\) as the sum of a vector in \(U\) and a vector in \(U^{\perp}\). a. \(\mathbf{v}=x^{2}, U=\operatorname{span}\\{x+1,9 x-5\\}\) b. \(\mathbf{v}=x^{2}+1, U=\operatorname{span}\\{1,2 x-1\\}\)

Short answer

For (a), \(\mathbf{v} = \text{proj}_U(\mathbf{v}) + \mathbf{v}_\perp\). For (b), split \(\mathbf{v} = \text{proj}_U(\mathbf{v}) + \mathbf{v}_\perp\).

Step-by-step solution

Identify basis and compute inner products

**Identify the basis for the subspaces**For part (a), the basis for \(U\) is \(\{x+1, 9x-5\}\). For part (b), the basis for \(U\) is \(\{1, 2x-1\}\).Compute the inner products needed for Gram-Schmidt orthogonalization and projection. Calculate \(\langle p, q \rangle = \int_{0}^{1} p(x)q(x)\, dx\) for combinations between \(\mathbf{v}\) and each basis vector.

Orthogonalize the basis using Gram-Schmidt process

**Apply Gram-Schmidt to create an orthogonal basis for \(U\)**For each basis vector \(p_i\), subtract the projection onto all previous orthogonal basis vectors to orthogonalize it. The formula for projection of \(q\) onto \(p\) is:\[\text{proj}_{p}(q) = \frac{\langle q, p \rangle}{\langle p, p \rangle} p\]Perform these calculations for both part (a) and part (b).

Find projection of \(\mathbf{v}\) onto \(U\)

**Project \(\mathbf{v}\) onto the orthogonal basis of \(U\)**Use the orthogonal basis to find the projection of \(\mathbf{v}\) onto \(U\):\[\text{proj}_{U}(\mathbf{v}) = \sum \frac{\langle \mathbf{v}, u_i \rangle}{\langle u_i, u_i \rangle} u_i\]Perform this calculation using the orthogonal basis vectors found in Step 2 for both parts (a) and (b).

Compute \(\mathbf{v} - \text{proj}_U(\mathbf{v})\)

**Calculate the component in \(U^\perp\)**The component of \(\mathbf{v}\) in \(U^\perp\) is found by subtracting the projection from \(\mathbf{v}\):\[\text{component in } U^\perp = \mathbf{v} - \text{proj}_U(\mathbf{v})\]Compute this for both parts (a) and (b) using the projection results from Step 3.

Key concepts

Inner Product

The concept of an inner product is crucial in linear algebra as it allows us to define angles and lengths in vector spaces. In this exercise, the inner product is defined for the polynomial space \( \mathbf{P}_2 \) by the integral \( \langle p, q \rangle = \int_{0}^{1} p(x) q(x) \, dx \). This integral evaluates the sum of the products of the polynomial coefficients over the interval from 0 to 1.

Inner products help identify orthogonal vectors, which are vectors that, when inserted into the inner product, give a result of zero. In this exercise, the inner product is used to calculate orthogonal projections and to differentiate between components of a vector within a subspace and its orthogonal complement.

• Ensures vectors have real functional spaces.
• Helps compute length and orthogonality.
• Facilitates calculations in the Gram-Schmidt process.

Orthogonal Projection

Orthogonal projection in linear algebra refers to projecting a vector onto a subspace such that the minimum distance exists between the vector and the subspace. This involves a crucial step to decompose any vector into two parts: one within the subspace (projection) and one orthogonal to the subspace.

The formula for the orthogonal projection \( \text{proj}_{U}(\mathbf{v}) \) is used here to describe projecting the vector \( \mathbf{v} \) onto a subspace \( U \). It reads as \[ \text{proj}_{U}(\mathbf{v}) = \sum \frac{\langle \mathbf{v}, u_i \rangle}{\langle u_i, u_i \rangle} u_i \]where \( u_i \) are the orthogonal basis vectors of the subspace \( U \).

Orthogonal projection allows us to understand how much of a vector lies in a given direction defined by a subspace and how much does not (resides in the orthogonal complement \( U^{\perp} \)). This aids in approximating data within a given subspace and analyzing components effectively.

Gram-Schmidt Process

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, which produces an orthogonal (sometimes orthonormal) set of vectors spanning the same subspace. It is essential for simplifying the structures within vector spaces.

In this exercise, the Gram-Schmidt process is applied to the basis of subspace \( U \) in order to form a set of orthogonal vectors. Suppose we have vectors \( \{ p_1, p_2 \} \), the process involves adjusting each vector sequentially by subtracting its projection onto the earlier vectors: \[ q_i = p_i - \sum_{j=1}^{i-1} \text{proj}_{q_j}(p_i) \] This ensures each \( q_i \) is orthogonal to all previous \( q_j \).

• Helps create an orthogonal basis making calculations easier.
• Improves numerical stability in calculations.
• Forms the basis for various algorithms in computational mathematics.

Vector Spaces

Vector spaces are fundamental structures in linear algebra comprising vectors which can be added together and multiplied by scalars, adhering to specific axioms. These spaces allow mathematicians and scientists to model and solve problems in multiple dimensions.

In the context of this exercise, we are dealing with the vectors in a polynomial space \( \mathbf{P}_2 \), forming vector spaces of functions defined over real numbers. Subspace \( U \) is spanned by sets of basis vectors, and the task involves writing vector \( \mathbf{v} \) as a sum of components that lie within this space and its orthogonal complement \( U^{\perp} \).

Understanding vector spaces ensures a deeper grasp on:

• How different dimensions interact and complement each other.
• Ways to manage multiple vectors in varied linear combinations.
• The behavior of complex calculus problems simplified through polynomial representations.