Problem 12 - Advanced Engineering Mathematics

Question

Determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 2,6,3\rangle, \mathbf{u}_{2}=\langle 1,-1,4\rangle, \mathbf{u}_{3}=\langle 3,2,1\rangle, \mathbf{u}_{4}=\langle 2,5,4\rangle $$

Step-by-Step Solution

Answer

The set of vectors is linearly dependent.

Step 1: Create a Matrix from Vectors

Form a matrix using the given vectors \( \mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4} \) as columns. The resulting matrix is \[ A = \begin{bmatrix} 2 & 1 & 3 & 2 \ 6 & -1 & 2 & 5 \ 3 & 4 & 1 & 4 \end{bmatrix} \].

Step 2: Convert the Matrix to Row Echelon Form

Use Gaussian elimination on matrix \( A \) to convert it into row echelon form. The goal is to simplify the matrix by performing row operations to make zeros below the pivot elements.

Step 3: Perform Row Operations

1. Subtract 3 times the first row from the third row to eliminate the 3 in the bottom-left corner.2. Subtract 3 times the first row from the second row to eliminate the 6 below the first pivot. After simplification, the matrix becomes: \[ \begin{bmatrix} 2 & 1 & 3 & 2 \ 0 & -4 & -7 & -1 \ 0 & 1 & -8 & 8 \end{bmatrix} \].

Step 4: Further Simplification

Add 4 times the second row to the third row to eliminate the 1 in the third row, second column:The matrix becomes:\[ \begin{bmatrix} 2 & 1 & 3 & 2 \ 0 & -4 & -7 & -1 \ 0 & 0 & -36 & 32 \end{bmatrix} \].

Step 5: Identify Pivots and Determine Dependence

Identify the pivot positions in the matrix. The pivots are the first non-zero elements from the left in each row: they are in columns 1, 2, and 3. Since there are fewer pivots (3) than vectors (4), the set of vectors is linearly dependent.

Key Concepts

Vectors

Vectors are mathematical entities within vector spaces that have both magnitude and direction. They are often represented in the form of tuples, like \( \mathbf{u} = \langle 2, 6, 3 \rangle \), where the numbers inside the angle brackets denote the vector's coordinates. These coordinates can correspond to respective axes in 2D or 3D spaces, depending on the context.

• Magnitudes: The length or size of the vector, calculated using the square root of the sum of the squares of its components.
• Direction: The vector's orientation in space, defined with respect to a coordinate system.

Vectors play a crucial role in various fields, such as physics and engineering, and are fundamental in studies involving linear algebra. This foundational knowledge helps determine more complex concepts like linear independence or dependence of vector sets.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. When applied to a matrix, this technique transforms it into a simpler form, known as row echelon form, through a sequence of row operations. These operations include:

• Row swapping: Interchanging two rows.
• Scaling: Multiplying a row by a non-zero constant.
• Row addition: Adding or subtracting a multiple of one row to another.

By using Gaussian elimination, we can better identify solutions to the system, determine the rank of the matrix, and check for the linear dependence of vectors. This process is integral in many areas of mathematics and computational fields, where simplification of complex systems is required.

Row Echelon Form

Row echelon form (REF) is a matrix form achieved through Gaussian elimination, aiding in understanding and solving systems of equations. A matrix is in REF when:

• All non-zero rows are above any rows of all zeros.
• The leading coefficient (or pivot) of a non-zero row is always to the right of the leading coefficient of the row above it.
• Each leading coefficient is the only non-zero entry in its column.

The goal of converting a matrix to row echelon form is to make it easier to identify solutions and understand the relationship between the dataset's vectors. This transformation simplifies the presence of zeros and highlights essential elements, making further computations more manageable.

Pivot Positions

Pivot positions are crucial in the context of matrices and Gaussian elimination. These positions identify the leading values within columns of a matrix once it is reduced to row echelon form. They represent the first non-zero entries from the left in each row. Recognizing these positions helps you understand several aspects:

• Rank of a Matrix: The number of pivots gives the matrix's rank, reflecting the maximum number of linearly independent rows/columns.
• Linear Independence: If the number of pivots equals the number of vectors, they are linearly independent. Otherwise, like in our exercise, the vectors are dependent.

Pivot positions guide us in judging the structural composition of the matrix. In cases where dependency is observed, it indicates some vectors can be expressed as a combination of others, providing insights into their interdependency.