Problem 14 - Advanced Engineering Mathematics

Question

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \begin{aligned} &\mathbf{u}_{1}=\langle 2,1,1,5\rangle, \mathbf{u}_{2}=\langle 2,2,1,1\rangle, \mathbf{u}_{3}=\langle 3,-1,6,1\rangle \\ &\mathbf{u}_{4}=\langle 1,1,1,-1\rangle \end{aligned} $$

Step-by-Step Solution

Answer

The set of vectors is linearly dependent.

Step 1: Set Up the System of Equations

To determine if the set of vectors \( \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \mathbf{u}_4\} \) is linearly independent, we set up the equation \( c_1 \mathbf{u}_1 + c_2 \mathbf{u}_2 + c_3 \mathbf{u}_3 + c_4 \mathbf{u}_4 = \mathbf{0} \), where \( \mathbf{0} \) is the zero vector in \( \mathbb{R}^4 \). This results in a system of equations for \( c_1, c_2, c_3, \) and \( c_4 \).

Step 2: Write the System in Matrix Form

Represent the vectors as rows in a matrix and form the augmented matrix: \[\begin{bmatrix}2 & 1 & 1 & 5 & | & 0 \2 & 2 & 1 & 1 & | & 0 \3 & -1 & 6 & 1 & | & 0 \1 & 1 & 1 & -1 & | & 0\end{bmatrix}\].

Step 3: Perform Row Reduction

Perform Gaussian elimination (row reduction) on the augmented matrix to find its row echelon form. The goal is to make the matrix below the main diagonal zero.

Step 4: Analyze Row-Reduced Form

After applying row reduction, the matrix should resemble a row echelon form. If you find at least one row that is entirely zeros (ignoring the last column), it implies that there are free variables, meaning the vectors are linearly dependent. If no row is entirely zero, the vectors are independent.

Step 5: Solve the System

If a zero row is found, solve the system of equations to find non-trivial solutions for the coefficients \( c_1, c_2, c_3, \) and \( c_4 \). A non-trivial solution confirms linear dependence.

Key Concepts

Vectors

Vectors are fundamental objects in mathematics and physics. They can be thought of as arrows pointing in space, having both direction and magnitude. In linear algebra, vectors facilitate the representation of shapes, transformations, and relations in higher dimensions.

• **Components**: Vectors are defined by their components. For instance, \( \mathbf{u}_1 = \langle 2, 1, 1, 5 \rangle \) is a vector in 4-dimensional space, \( \mathbb{R}^4 \).
• **Operations**: Vectors can be added together, scaled by a number (scalar multiplication), and used in complex operations like dot product and cross product.

They represent a point in space relative to an origin. Understanding vectors is essential for solving problems involving spatial relationships, such as determining if a set of vectors is linearly independent or dependent.

• **Basis of a Space**: In linear algebra, vectors are essential in forming the basis of a vector space. A basis is a linearly independent set of vectors that span the whole space, meaning any vector in the space can be represented as a combination of the basis vectors.

Linear Dependence

Linear dependence is a concept indicating that a set of vectors does not uniquely span a vector space. It describes a relationship where some vectors in a set can be expressed as a combination of others.

• **Definition**: A set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \) is linearly dependent if there are scalar coefficients, not all zero, such that the linear combination \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \ldots + c_n \mathbf{v}_n = \mathbf{0} \), where \( \mathbf{0} \) is the zero vector.
• **Implications**: This means at least one vector in the set can be written as a linear combination of the others.

Linear dependence demonstrates that the vectors do not add new dimensions to the space they occupy, causing redundancy.
Whenever a vector can be written as a combination of others, it is not adding any new information to the pattern they form. Tools like Gaussian elimination help uncover these relationships by simplifying the equations, revealing any dependencies among the vectors.

Row Reduction

Row reduction is a method used in linear algebra to simplify matrices and solve systems of linear equations. It is closely related to Gaussian elimination.

• **Purpose**: The main purpose of row reduction is to convert a matrix into a simpler form, often a row-echelon form or a reduced row-echelon form.
• **Process**: The process involves using elementary row operations to make the entries below the diagonal of a matrix zero.

These operations are:

• Swapping rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to/from another row

Performing row reduction efficiently reveals the structure of a system of equations. It makes checking for linear independence straightforward, as it helps identify if rows of the matrix become zero, indicating a linear dependence.
Moreover, it assists in solving the system by clearly showing if there are free variables, which is a telltale sign of dependency.