Problem 14 - Differential Equations and Linear Algebra

Question

Write the column vectors and row vectors of the given matrix. $$A=\left[\begin{array}{rrr} 1 & 3 & -4 \\ -1 & -2 & 5 \\ 2 & 6 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

Row vectors: 1. \( \begin{bmatrix} 1 & 3 & -4 \end{bmatrix} \) 2. \( \begin{bmatrix} -1 & -2 & 5 \end{bmatrix} \) 3. \( \begin{bmatrix} 2 & 6 & 7 \end{bmatrix} \) Column vectors: 1. \( \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \) 2. \( \begin{bmatrix} 3 \\ -2 \\ 6 \end{bmatrix} \) 3. \( \begin{bmatrix} -4 \\ 5 \\ 7 \end{bmatrix} \)

Step 1: Identifying row vectors

First, we will look at the given matrix and identify its rows. The matrix has three rows: 1. Row 1: [1, 3, -4] 2. Row 2: [-1, -2, 5] 3. Row 3: [2, 6, 7] These will be our row vectors.

Step 2: Identifying column vectors

Next, we will identify the columns of the matrix. The matrix has three columns: 1. Column 1: [1, -1, 2] 2. Column 2: [3, -2, 6] 3. Column 3: [-4, 5, 7] These will be our column vectors.

Step 3: Writing the results

Now that we have identified the row vectors and column vectors of the matrix, we will write them down: Row vectors: 1. \( \begin{bmatrix} 1 & 3 & -4 \end{bmatrix} \) 2. \( \begin{bmatrix} -1 & -2 & 5 \end{bmatrix} \) 3. \( \begin{bmatrix} 2 & 6 & 7 \end{bmatrix} \) Column vectors: 1. \( \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} \) 2. \( \begin{bmatrix} 3 \\ -2 \\ 6 \end{bmatrix} \) 3. \( \begin{bmatrix} -4 \\ 5 \\ 7 \end{bmatrix} \)

Key Concepts

Row Vectors

Row vectors are simply rows taken from a given matrix. Each row vector is a horizontal array of elements forming a single row. In the matrix given in the exercise, the matrix \( A \) is:

\[A = \begin{bmatrix} 1 & 3 & -4 \ -1 & -2 & 5 \ 2 & 6 & 7 \end{bmatrix}\]Each of the three rows is a separate row vector:

• Row 1: \( \begin{bmatrix} 1 & 3 & -4 \end{bmatrix} \)
• Row 2: \( \begin{bmatrix} -1 & -2 & 5 \end{bmatrix} \)
• Row 3: \( \begin{bmatrix} 2 & 6 & 7 \end{bmatrix} \)

Row vectors play an essential role in many matrix operations, like matrix addition and scalar multiplication.
They are also crucial for understanding concepts like row echelon form in solving linear systems.

Column Vectors

Column vectors are the vertical slices of a matrix. Each column vector is a vertical array of numbers, representing a single column. In the matrix \( A \), each column vector can be seen clearly when the matrix is deconstructed:

\[A = \begin{bmatrix} 1 & 3 & -4 \ -1 & -2 & 5 \ 2 & 6 & 7 \end{bmatrix}\]The columns are:

• Column 1: \( \begin{bmatrix} 1 \ -1 \ 2 \end{bmatrix} \)
• Column 2: \( \begin{bmatrix} 3 \ -2 \ 6 \end{bmatrix} \)
• Column 3: \( \begin{bmatrix} -4 \ 5 \ 7 \end{bmatrix} \)

Column vectors are critical in linear algebra for operations like matrix multiplication.
They define the range or column space of a matrix which is key in understanding transformations and other applications.

Linear Algebra

Linear algebra is a field of mathematics that studies vectors, matrices, and linear transformations. It's a foundational tool for industries like data science, engineering, and physics. At the heart of linear algebra are two main operations: vector addition and scalar multiplication, both of which help define various mathematical structures.

In linear algebra, matrices represent linear transformations that map vectors from one space to another. The matrix's rows and columns play a role in representing various vectors and bases. This makes operations involving row vectors and column vectors, like creating new matrices through addition, essential in linear algebra.

• Understanding these helps in dimensional analysis and finding solutions to systems of linear equations.
• Concepts like vector spaces, eigenvalues, and eigenvectors further build on the basic understanding of vectors and matrices.

This makes knowledge of row and column vectors vital for anyone studying or working in disciplines that involve linear algebra.

Matrix Identification

Matrix identification revolves around recognizing and understanding the different types of matrices based on their arrangement and purpose. A matrix can be characterized by its size, depicted as \( m \times n \) (where \( m \) is the number of rows and \( n \) is the number of columns).

In the exercise matrix \( A \), which is a \( 3 \times 3 \) matrix, this means there are three row vectors and three column vectors. Identifying matrices correctly helps determine what kinds of operations can be performed. Matrices might also be classified based on features like:

• Square matrix: where the number of rows equals the number of columns
• Zero matrix: where all elements are zero
• Identity matrix: a special kind of square matrix with \( 1\)s along the diagonal

Understanding matrix identification is crucial for learning how different types affect computations, such as solving equations or performing transformations in linear algebra.