Question

a. Show that \(\mathbf{x} \cdot \mathbf{y}=\frac{1}{4}\left[\|\mathbf{x}+\mathbf{y}\|^{2}-\|\mathbf{x}-\mathbf{y}\|^{2}\right]\) for all \(\mathbf{x}\), \(\mathbf{y}\) in \(\mathbb{R}^{n}\) b. Show that \(\|\mathbf{x}\|^{2}+\|\mathbf{y}\|^{2}=\frac{1}{2}\left[\|\mathbf{x}+\mathbf{y}\|^{2}+\|\mathbf{x}-\mathbf{y}\|^{2}\right]\) for all \(\mathbf{x}, \mathbf{y}\) in \(\mathbb{R}^{n}\)

Short answer

Part (a): Expand and subtract norms; Part (b): Expand and add norms.

Step-by-step solution

Expand the Norms for Part (a)

To prove the given equation in part (a), start by expanding both squared norm terms.The norm squared of a vector \( \mathbf{a} \) in \( \mathbb{R}^{n} \) is given by: \[ \| \mathbf{a} \|^2 = \mathbf{a} \cdot \mathbf{a} \].Thus, for the vector sum \( \mathbf{x} + \mathbf{y} \), we have:\[ \| \mathbf{x} + \mathbf{y} \|^2 = (\mathbf{x} + \mathbf{y}) \cdot (\mathbf{x} + \mathbf{y}) = \mathbf{x} \cdot \mathbf{x} + 2 \mathbf{x} \cdot \mathbf{y} + \mathbf{y} \cdot \mathbf{y} = \| \mathbf{x} \|^2 + 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2 \].For the vector difference \( \mathbf{x} - \mathbf{y} \), we have:\[ \| \mathbf{x} - \mathbf{y} \|^2 = (\mathbf{x} - \mathbf{y}) \cdot (\mathbf{x} - \mathbf{y}) = \mathbf{x} \cdot \mathbf{x} - 2 \mathbf{x} \cdot \mathbf{y} + \mathbf{y} \cdot \mathbf{y} = \| \mathbf{x} \|^2 - 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2 \].

Subtract the Expanded Norms for Part (a)

Now, subtract the squared norm of \( \mathbf{x} - \mathbf{y} \) from the squared norm of \( \mathbf{x} + \mathbf{y} \):\[ \| \mathbf{x} + \mathbf{y} \|^2 - \| \mathbf{x} - \mathbf{y} \|^2 = (\| \mathbf{x} \|^2 + 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2) - (\| \mathbf{x} \|^2 - 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2) \].Simplify the expression:\[ = 4 \mathbf{x} \cdot \mathbf{y} \].Thus, \[ \mathbf{x} \cdot \mathbf{y} = \frac{1}{4} (\| \mathbf{x} + \mathbf{y} \|^2 - \| \mathbf{x} - \mathbf{y} \|^2) \].This completes part (a).

Expand the Norms for Part (b)

To prove the statement in part (b), use the same formulae for the norms already developed:Recall:\[ \| \mathbf{x} + \mathbf{y} \|^2 = \| \mathbf{x} \|^2 + 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2 \].\[ \| \mathbf{x} - \mathbf{y} \|^2 = \| \mathbf{x} \|^2 - 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2 \].

Add the Expanded Norms for Part (b)

Add the two expanded expressions to prove part (b):\[ \| \mathbf{x} + \mathbf{y} \|^2 + \| \mathbf{x} - \mathbf{y} \|^2 = (\| \mathbf{x} \|^2 + 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2) + (\| \mathbf{x} \|^2 - 2 \mathbf{x} \cdot \mathbf{y} + \| \mathbf{y} \|^2) \].Simplify the expression:\[ = 2 \| \mathbf{x} \|^2 + 2 \| \mathbf{y} \|^2 \].Dividing by 2, we obtain:\[ \| \mathbf{x} \|^2 + \| \mathbf{y} \|^2 = \frac{1}{2} (\| \mathbf{x} + \mathbf{y} \|^2 + \| \mathbf{x} - \mathbf{y} \|^2) \].This completes part (b).

Key concepts

Vector Norms

A vector norm is a way to measure the length or magnitude of a vector. Imagine it as how far a point is from the origin in a numeric space. For a vector \( \mathbf{a} \) in \( \mathbb{R}^n \), the norm is what you get when you add up the squares of each component of the vector, then take the square root. But often we deal with the squared norm, especially in proofs like these, which is simply the sum of the squares of its components without taking the square root. The squared norm of a vector \( \mathbf{a} \) is represented as \( \| \mathbf{a} \|^2 = \mathbf{a} \cdot \mathbf{a} \). This is also equal to the dot product of the vector with itself.

Whenever you see norm squared in mathematical problems or proofs, think of it as calculating the length squared of that vector, a simpler form which is frequently used in analytics and formulation of proofs due to its ease of manipulation.

Vector Addition and Subtraction

Adding or subtracting vectors is akin to combining or separating their paths. Think about vectors as arrows in space. When you add two vectors \( \mathbf{x} \) and \( \mathbf{y} \), you align them tail to head, essentially forming a new vector by following both paths in sequence. This results in \( \mathbf{x} + \mathbf{y} \).

Subtracting vectors \( (\mathbf{x} - \mathbf{y}) \) can be visualized by adding the negative of one vector to another. So \( \mathbf{x} - \mathbf{y} \) means you take the path of \( \mathbf{x} \) and then cancel the path of \( \mathbf{y} \).

These operations are crucial because they affect the overall direction and magnitude, reflecting on how we expand and simplify vector norms in proofs. In the given exercise, vector addition and subtraction help to break down complex expressions into manageable parts that can be expanded and simplified using their dot product properties.

Mathematical Proofs

Mathematical proofs are like a structured argument that demonstrate why a particular mathematical statement is true. They are crucial in establishing facts that are accepted by the mathematical community. Proofs start with known facts—like the properties of vector addition and subtraction—and use logic to arrive at the statement you want to prove.

In the exercise, two proofs are demonstrated. The first follows from expanding vector norms, leveraging the property \( \| \mathbf{x} + \mathbf{y} \|^2 - \| \mathbf{x} - \mathbf{y} \|^2 \) to demonstrate the dot product formula. Notably, each step connects back to the properties of the vector dot product, showing consistency in mathematical reasoning.

The second part uses addition of expanded norms to equally split into known components. By proving \( \| \mathbf{x} \|^2 + \| \mathbf{y} \|^2 \) equals a half of two separate conditions, it reinforces the logic that every derived fact has a basis on initial axioms and properties, such as those of vector spaces. Mathematical proofs help build the foundation of reliability for equations and solutions in higher mathematics.