Problem 13 - Advanced Engineering Mathematics

Question

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 1,-1,3,-1\rangle, \mathbf{u}_{2}=\langle 1,-1,4,2\rangle, \mathbf{u}_{3}=\langle 1,-1,5,7\rangle $$

Step-by-Step Solution

Answer

The vectors are linearly independent.

Step 1: Formulate the Matrix

Consider the vectors as the rows of a matrix. The given vectors are \( \mathbf{u}_1 = \langle 1, -1, 3, -1 \rangle \), \( \mathbf{u}_2 = \langle 1, -1, 4, 2 \rangle \), and \( \mathbf{u}_3 = \langle 1, -1, 5, 7 \rangle \). Formulate a matrix \( A \) with these vectors as its rows:\[A = \begin{bmatrix}1 & -1 & 3 & -1 \1 & -1 & 4 & 2 \1 & -1 & 5 & 7\end{bmatrix}\]

Step 2: Determine Row Echelon Form (REF)

Perform row operations to reduce the matrix to the Row Echelon Form (REF) to analyze its linear independence. Begin by subtracting the first row from the second row and from the third row:Second row operation: \( R_2 - R_1 \rightarrow R_2 \)Third row operation: \( R_3 - R_1 \rightarrow R_3 \)\[\begin{bmatrix}1 & -1 & 3 & -1 \0 & 0 & 1 & 3 \0 & 0 & 2 & 8\end{bmatrix}\]

Step 3: Further Simplification

Continue reducing by eliminating the third row element in the third column using the second row:Third row operation: \( R_3 - 2R_2 \rightarrow R_3 \)\[\begin{bmatrix}1 & -1 & 3 & -1 \0 & 0 & 1 & 3 \0 & 0 & 0 & 2\end{bmatrix}\]

Step 4: Analyze REF for Pivot Positions

Check the reduced row echelon form for pivot positions. There are three non-zero rows, indicating a pivot position in each row: column 1, 3, and 4. This implies the matrix has full row rank, meaning the vectors are linearly independent.

Key Concepts

Row Echelon Form

Understanding the row echelon form of a matrix is crucial in determining linear independence. The row echelon form is a simplified version of a matrix that is derived through elementary row operations. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row to another.

This form has a staircase appearance. Specifically, each leading entry is to the right of the leading entry in the previous row, and all entries below the leading entries are zeros. By reducing a matrix to its row echelon form, you can easily observe the structure and connectivity of the vectors across the rows.

• This helps to pinpoint which rows (vectors) contribute independently to the span of the space.
• In our exercise, the transformation of the matrix to its row echelon form allowed us to identify the pivot positions, crucial in verifying linear independence.

Pivot Positions

Pivot positions are vital in the study of matrices and vector spaces. These are the first non-zero entries in each row of a matrix that has been transformed into row echelon form. These positions clarify the structural independence of vectors. In a simpler sense, a pivot in a column indicates a key contributor to the matrix's rank and linear independence between the vectors.

Understanding pivot positions can guide us in determining whether the entire set of rows (or columns) is linearly independent. If each row contains a pivot position, it means that each vector adds new information about the vector space it spans.

• In our exercise, after reducing the matrix to its simplified form, we identified pivots in columns 1, 3, and 4.
• This confirmed that the provided vectors were linearly independent, as each row had a unique pivot position.

Vector Spaces

Vector spaces are fundamental concepts in linear algebra that provide a fascinating insight into the world of mathematics. A vector space is a collection of vectors that can be added together and multiplied by scalars. This concept is central to understanding how vectors relate to and influence each other.

In our exercise involving determining linear independence, the vector space was implicitly defined by the capacity of the given set of vectors to span or cover a specific space.

• When vectors are linearly independent, they uniquely contribute to the vector space, meaning no vector is simply a combination of the others.
• This property is crucial in applications like solving systems of equations, where determining a unique solution is equivalent to having independent vectors.

By examining the vector space, one can appreciate how these sets of vectors interact and define the dimensions and limits of mathematical constructs.