Problem 3 - Differential Equations and Linear Algebra

Question

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(2,-1),(3,2),(0,1)\\}$$.

Step-by-Step Solution

Answer

The given set of vectors is linearly independent in \(\mathbb{R}^{n}\) as the row echelon form of the matrix shows that the system has a unique solution.

Step 1: Write down the linear system

Use the given set of vectors as the columns of a matrix and set up a homogeneous linear system: \[ \begin{bmatrix} 2 & 3 & 0 \\ -1 & 2 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \]

Step 2: Find the row echelon form of the matrix

Perform elementary row operations on the matrix to find its row echelon form: \[ \begin{bmatrix} 2 & 3 & 0 \\ -1 & 2 & 1 \end{bmatrix} \underset{(1)\leftrightarrow(-\frac{1}{2} R_1+R_2)}{\rightarrow} \begin{bmatrix} 2 & 3 & 0 \\ 0 & \frac{7}{2} & 1 \end{bmatrix}. \]

Step 3: Determine if there is a unique solution or not

The row echelon form of the matrix shows that the system has a unique solution since there are no rows that are entirely composed of zeros. Therefore, the vectors are linearly independent.

Step 4: Conclude the results

The given set of vectors is linearly independent in \(\mathbb{R}^{n}\). Since the vectors are linearly independent, there is no need to find a dependency relationship.

Key Concepts

Vector Spaces

In mathematics, a vector space is a crucial concept in linear algebra. It's simply a collection of vectors where you can perform addition and scalar multiplication. For example, the space \(\mathbb{R}^n\) consists of all vectors with \(n\) real numbers.
These vectors can be added together and multiplied by real numbers, creating other vectors in the same space.

• Vectors: These are ordered lists of numbers.
• Addition: You can add two vectors to get another vector.
• Scalar Multiplication: You can multiply a vector by a number to scale it.

Understanding vector spaces helps in analyzing systems of linear equations and transformations.

Homogeneous System

A homogeneous system is a type of linear system where all constant terms are zero. That means the system of equations can be expressed in matrix form as \(A\mathbf{x} = \mathbf{0}\). The example in the exercise focuses on such a system:

• Matrix Representation: The vectors form the columns of a matrix.
• Solution: The trivial solution is always \(x_1 = x_2 = x_3 = 0\).
• Non-trivial Solutions: If they exist, the vectors are linearly dependent.

Determining whether solutions beside the trivial one exist is key to checking linear independence.

Row Echelon Form

To determine the linear independence of vectors, you convert the matrix into row echelon form using Gaussian elimination. This focused approach transforms the system into a more understandable structure:

• Leading Coefficients: Each non-zero row starts with a leading 1.
• Zero Rows: All zero rows are at the bottom.
• Staircase Pattern: Leading coefficients must move down and to the right.

The presence of a unique solution, as shown in the example, indicates linear independence.

Elementary Row Operations

These operations are used to manipulate matrices while solving a system of linear equations. They're tools to transform matrices into simpler forms such as row echelon form.

• Row Swapping: You can interchange two rows.
• Scalar Multiplication: Multiply a row by a non-zero number.
• Row Addition: Add or subtract one row from another.

These efficient operations preserve the solution set and help reveal the nature of vector dependencies.