Problem 1 - Differential Equations and Linear Algebra

Question

Let \(r\) and \(s\) denote scalars and let \(\mathbf{v}\) and \(\mathbf{w}\) denote vectors in \(\mathbb{R}^{5}\). $$\text { Prove that }(r+s) \mathbf{v}=r \mathbf{v}+s \mathbf{v}$$

Step-by-Step Solution

Answer

In the vector space \(\mathbb{R}^5\), to prove that \((r+s) \mathbf{v} = r \mathbf{v} + s \mathbf{v}\), we first evaluated the left-hand side \((r+s)\mathbf{v} = \begin{pmatrix} (r+s) v_1 \\ (r+s) v_2 \\ (r+s) v_3 \\ (r+s) v_4 \\ (r+s) v_5 \end{pmatrix}\), and the right-hand side \(r\mathbf{v} + s\mathbf{v} = \begin{pmatrix} (r v_1 + s v_1) \\ (r v_2 + s v_2) \\ (r v_3 + s v_3) \\ (r v_4 + s v_4) \\ (r v_5 + s v_5) \end{pmatrix}\). Both sides are equal component-wise, thus the equality holds as proved.

Step 1: Understand Scalar-Vector Multiplication

Scalar-vector multiplication, in general, refers to the multiplication of a scalar (a single real number) by a vector. In this case, our vector space is \(\mathbb{R}^{5}\), which means each vector \(\mathbf{v}\) has five components: $$\mathbf{v}=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \\ v_5 \end{pmatrix}$$. When a scalar \(r\) is multiplied with the vector \(\mathbf{v}\), each component of the vector is multiplied by the scalar: $$r\mathbf{v}=\begin{pmatrix} r v_1 \\ r v_2 \\ r v_3 \\ r v_4 \\ r v_5 \end{pmatrix}$$.

Step 2: Calculate the Left-Hand Side of the Equality

We need to evaluate the left-hand side of the given equation: \((r+s) \mathbf{v}\). To do this, we first add the scalars \(r\) and \(s\), and then multiply the sum by the vector \(\mathbf{v}\). So, for each component, the result would be: $$(r+s)\mathbf{v} = \begin{pmatrix} (r+s) v_1 \\ (r+s) v_2 \\ (r+s) v_3 \\ (r+s) v_4 \\ (r+s) v_5 \end{pmatrix}$$.

Step 3: Calculate the Right-Hand Side of the Equality

Now, we need to compute the right-hand side of the equation, which is \(r \mathbf{v} + s \mathbf{v}\). We will first find \(r \mathbf{v}\) and \(s \mathbf{v}\), then add them component-wise. Thus, we get: $$r\mathbf{v} = \begin{pmatrix} r v_1 \\ r v_2 \\ r v_3 \\ r v_4 \\ r v_5 \end{pmatrix}$$ and $$s\mathbf{v} = \begin{pmatrix} s v_1 \\ s v_2 \\ s v_3 \\ s v_4 \\ s v_5 \end{pmatrix}$$. Adding them together, we obtain: $$r \mathbf{v} + s\mathbf{v} = \begin{pmatrix} (r v_1 + s v_1) \\ (r v_2 + s v_2) \\ (r v_3 + s v_3) \\ (r v_4 + s v_4) \\ (r v_5 + s v_5) \end{pmatrix}$$.

Step 4: Compare the Left-Hand Side and Right-Hand Side

Now, we will compare the results from Steps 2 and 3. From our calculations, we have: $$(r+s) \mathbf{v} = \begin{pmatrix} (r+s) v_1 \\ (r+s) v_2 \\ (r+s) v_3 \\ (r+s) v_4 \\ (r+s) v_5 \end{pmatrix}$$ and $$r \mathbf{v} + s \mathbf{v} = \begin{pmatrix} (r v_1 + s v_1) \\ (r v_2 + s v_2) \\ (r v_3 + s v_3) \\ (r v_4 + s v_4) \\ (r v_5 + s v_5) \end{pmatrix}$$. Both sides are equal component-wise since for all components \(i\), we have \((r+s)v_i = r v_i + s v_i\). Thus, the equality holds: $$(r+s) \mathbf{v}=r \mathbf{v}+s \mathbf{v}$$ In conclusion, we have proved that the distributive property of scalar-vector multiplication holds for the given vectors and scalars in \(\mathbb{R}^{5}\).

Key Concepts

Scalar-Vector Multiplication

Scalar-vector multiplication is a fundamental concept in linear algebra. It involves multiplying a scalar, which is just a regular number, with a vector. A vector in a vector space like \( \mathbb{R}^5 \) is essentially a list of numbers, each representing a component of the vector. For instance, a vector \( \mathbf{v} \) in \( \mathbb{R}^5 \) can be expressed as \( \mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \ v_4 \ v_5 \end{pmatrix} \).

When performing scalar-vector multiplication, each component of the vector is multiplied by the scalar. For example, if we have a scalar \( r \) and the vector \( \mathbf{v} \), the product \( r\mathbf{v} \) becomes \( \begin{pmatrix} rv_1 \ rv_2 \ rv_3 \ rv_4 \ rv_5 \end{pmatrix} \).

This operation is crucial because it helps in scaling vectors, which is necessary in various applications, from physics simulations to graphics transformations.

Vector Space

A vector space is a set of vectors where you can perform two operations: vector addition and scalar multiplication. It's a key structure in linear algebra, allowing for the development of many mathematical models.

In a vector space like \( \mathbb{R}^5 \), vectors such as \( \mathbf{v} \) and \( \mathbf{w} \) can be added or scaled by scalars. This flexibility makes vector spaces powerful tools for describing real-world phenomena.

A vector space adheres to specific rules, called axioms, that include:

• Closure under vector addition and scalar multiplication.
• The existence of a zero vector, which is an additive identity.
• The presence of additive inverses for every vector.

These properties ensure that operations within the vector space remain consistent and predictable, facilitating robust mathematical modeling and problem-solving.

Component-Wise Addition

Component-wise addition is a straightforward process used extensively in computing and mathematics. It involves adding corresponding components of two vectors to form a new vector. Consider two vectors \( \mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \ v_4 \ v_5 \end{pmatrix} \) and \( \mathbf{w} = \begin{pmatrix} w_1 \ w_2 \ w_3 \ w_4 \ w_5 \end{pmatrix} \). The component-wise addition \( \mathbf{v} + \mathbf{w} \) results in a new vector \( \begin{pmatrix} v_1 + w_1 \ v_2 + w_2 \ v_3 + w_3 \ v_4 + w_4 \ v_5 + w_5 \end{pmatrix} \).

This technique is essential because it maintains the structure of mathematical operations in a vector space. It ensures that the vector space remains closed under addition, which means the sum of any two vectors in the space produces another vector within the same space. Component-wise addition is intuitive and simplifies computations, making it a valuable method in various fields like data analysis and engineering.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vectors, vector spaces, and linear transformations. It's foundational for many scientific and engineering disciplines. The study of matrices and vector operations such as addition, scalar multiplication, and the application of the distributive property are all part of linear algebra.

In this context, the distributive property is an essential concept, allowing us to take operations such as addition of scalars into and factor out of vector multiplication. For example, in the property \( (r+s) \mathbf{v} = r \mathbf{v} + s \mathbf{v} \), it showcases the relationship between mathematics and its logical syntax.

Linear algebra provides tools for solving systems of equations, transforming geometric shapes, and analyzing data sets. It enables the modeling of complex systems by simplifying them into forms that are easier to analyze. With applications in data science, machine learning, and even computer graphics, studying linear algebra opens the door to understanding and exploiting the mathematical structure of the world.