Problem 14 - Advanced Engineering Mathematics

Question

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 given set of vectors is linearly independent.

Step 1: Set Up the Problem

We are given four vectors \( \mathbf{u}_{1}=\langle 2,1,1,5 \rangle \), \( \mathbf{u}_{2}=\langle 2,2,1,1 \rangle \), \( \mathbf{u}_{3}=\langle 3,-1,6,1 \rangle \), and \( \mathbf{u}_{4}=\langle 1,1,1,-1 \rangle \). We need to determine if they are linearly dependent or independent by setting up an equation for linear dependence: \( c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + c_3\mathbf{u}_3 + c_4\mathbf{u}_4 = \mathbf{0} \).

Step 2: Write the System of Equations

Translate the equation \( c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + c_3\mathbf{u}_3 + c_4\mathbf{u}_4 = \mathbf{0} \) into a system of linear equations: \[\begin{align*}2c_1 + 2c_2 + 3c_3 + c_4 &= 0, \c_1 + 2c_2 - c_3 + c_4 &= 0, \c_1 + c_2 + 6c_3 + c_4 &= 0, \5c_1 + c_2 + c_3 - c_4 &= 0.\end{align*}\]

Step 3: Form the Augmented Matrix

Create an augmented matrix from the above set of equations:\[\begin{bmatrix}2 & 2 & 3 & 1 & | 0 \1 & 2 & -1 & 1 & | 0 \1 & 1 & 6 & 1 & | 0 \5 & 1 & 1 & -1 & | 0 \end{bmatrix}\]

Step 4: Row Reduce the Matrix

Use Gaussian elimination (or row reduction) to reduce the matrix to row echelon form or reduced row echelon form:\[\begin{bmatrix}1 & 0 & 0 & 0 & | 0 \0 & 1 & 0 & 0 & | 0 \0 & 0 & 1 & 0 & | 0 \0 & 0 & 0 & 1 & | 0 \end{bmatrix}\]

Step 5: Analyze the Row Reduction

Each variable has a leading 1 in its column with no free variables. This indicates that the only solution to the system corresponds to \( c_1 = c_2 = c_3 = c_4 = 0 \).

Step 6: Conclusion

Since the only solution is the trivial solution, the vectors \( \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \mathbf{u}_4 \) are linearly independent.

Key Concepts

Vectors

Vectors are essential building blocks in mathematics and physics. They are geometric objects that have both a magnitude and a direction. In linear algebra, vectors are often represented as ordered lists of numbers which define their position in space.

• In an n-dimensional space, a vector can be denoted as \((v_1, v_2, ..., v_n)\), where each \(v_i\) represents a component of the vector in that dimension.
• Vectors can be added together and scaled by numbers, which are known as scalars.
• A set of vectors is linearly independent if no vector in the set can be written as a combination of the others.

These concepts are crucial when dealing with problems involving vector spaces and linear transformations.

Linear Algebra

Linear algebra is a branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of vectors, vector spaces, and systems of linear equations.
Its most fundamental tasks involve computing vector collections, solving systems of linear equations, and working with matrix transformations.

• The main goal of linear algebra is to provide a concise and easy-to-understand framework for solving complex geometry and physics problems.
• Key concepts include matrices, determinants, eigenvalues, and eigenvectors.

Understanding linear algebra is essential for solving problems where multiple variables or unknowns are involved.

Row Reduction

Row reduction is a systematic method of simplifying matrices to determine solutions for systems of linear equations through row operations.
This process converts a matrix into a simpler form, such as row echelon form or reduced row echelon form, aiding in finding solutions easily.

• Row operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows from each other.
• The goal is to obtain a matrix where the process of back-substitution is straightforward.

Using row reduction, you can test for linear dependence by observing whether the matrix row-reduces to a form with non-zero entries across all columns, indicating a unique solution (linear independence).

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations and to calculate determinants and inverses of matrices.
It involves performing a sequence of operations on the augmented matrix to transform it into a row-echelon form.

• The process allows for determining the rank of the matrix and subsequently the number of solutions to the corresponding system of equations.
• If the matrix can be reduced to a form where each variable corresponds to a leading entry (a non-zero pivot), it implies all solutions are trivial, confirming the set's linear independence.

This technique is foundational in linear algebra since it underpins the solution of linear systems and the decomposition of matrices.