Problem 17 - Advanced Engineering Mathematics

Question

Let \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\) be the first, second, and third column vectors, respectively, of the matrix $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 1 & 7 \\ 1 & 0 & 2 \\ -1 & 5 & 13 \end{array}\right) $$ What can we conclude about \(\operatorname{rank}(\mathbf{A})\) from the observation \(2 \mathbf{v}_{1}+3 \mathbf{v}_{2}-\mathbf{v}_{3}=\mathbf{0} ?\) [Hint: Read the Remarks at the end of this section.]

Step-by-Step Solution

Answer

The rank of \(\mathbf{A}\) is 2.

Step 1: Understand Matrix Columns

First, identify the column vectors \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) of the matrix \(\mathbf{A}\). These are:\[ \mathbf{v}_{1} = \begin{pmatrix} 2 \ 1 \ -1 \end{pmatrix}, \quad \mathbf{v}_{2} = \begin{pmatrix} 1 \ 0 \ 5 \end{pmatrix}, \quad \mathbf{v}_{3} = \begin{pmatrix} 7 \ 2 \ 13 \end{pmatrix} \]

Step 2: Interpret the Given Equation

The equation given is \(2 \mathbf{v}_{1} + 3 \mathbf{v}_{2} - \mathbf{v}_{3} = \mathbf{0}\). This equation shows a linear dependency among the columns of \(\mathbf{A}\). It indicates there exists a nontrivial linear combination of the columns that results in the zero vector.

Step 3: Determine Linear Dependence

Since \(2 \mathbf{v}_{1} + 3 \mathbf{v}_{2} - \mathbf{v}_{3} = \mathbf{0}\) holds, \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) are linearly dependent. In a set of three vectors, if they are linearly dependent, the rank of the matrix formed by these vectors (\(\mathbf{A}\)) is less than 3.

Step 4: Conclude the Rank

Since the maximum number of linearly independent vectors is less than 3 (because they are dependent), the rank of \(\mathbf{A}\) must be 2. This is the largest number of linearly independent columns.

Key Concepts

Understanding Linear Dependence

Linear dependence is a concept that helps us understand the relationship between different vectors. When we say vectors are linearly dependent, it means there is a way to combine them to get the zero vector without all the coefficients being zero.
In simpler terms, if you can add, subtract, or scale the vectors and end up with zero, they are dependent. In our case, the equation \(2 \mathbf{v}_{1} + 3 \mathbf{v}_{2} - \mathbf{v}_{3} = \mathbf{0}\) shows that these three vectors are dependent.
Why is this important? Linear dependence means some of the vectors don't provide new direction or dimension—in essence, they repeat the information captured by other vectors.

• If you find one such relationship, it implies that the vectors “overlap” in a certain sense.
• Not all vectors point toward unique new directions.
• A clue to understanding the rank of a matrix—the number of linearly independent vectors.

This understanding helps assess that the space vectors span isn't as full-rich as it could be if all were independent.

The Role of Column Vectors

Column vectors make up the columns of a matrix. They are essential because they represent directions or axes in space regarding how a system behaves or is structured.
Each column vector in a matrix like \( \mathbf{A} \) in the problem can be thought of as an individual player, each representing one aspect of the collective system the matrix describes.
In three-dimensional space, having all independent vectors could mean each points towards a different dimension.

• These vectors can be visualized as arrows or directed lines originating from the origin.
• Each column vector here, \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \), are parts of the matrix \( \mathbf{A} \).
• They provide the "building blocks" for describing what the matrix achieves when applied to other vectors.

Their importance lies in their ability to show what kind of dimensions or directions are captured within another space represented by the matrix.

Decoding Linear Combinations

Linear combinations are at the heart of understanding any system expressed as a set of equations or vectors.
Generally, a linear combination of vectors is formed by multiplying each vector by a number (called a scalar), and then adding the results together. In the exercise, the linear combination \(2 \mathbf{v}_1 + 3 \mathbf{v}_2 - \mathbf{v}_3 = \mathbf{0}\) results in the zero vector.
This shows us that such combinations aren't adding up to create new vectors; instead, they circle back to zero. When applied:

• A linear combination can help check if vectors are dependent or independent.
• They are used to express solutions or transformations within vector spaces.
• Finding such combinations that equate to zero reveals the potential rank of matrices like \( \mathbf{A} \).

By understanding these combinations, students can unravel the heart of many linear algebra problems, grasping what's possible or constrained within the framework given by the matrix.