Problem 15 - Differential Equations and Linear Algebra

Question

Write the column vectors and row vectors of the given matrix. $$A=\left[\begin{array}{ccc} 2 & 10 & 6 \\ 5 & -1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The row vectors of the matrix are: \(R_1 = \left[\begin{array}{ccc} 2 & 10 & 6 \end{array}\right]\) and \(R_2 = \left[\begin{array}{ccc} 5 & -1 & 3 \end{array}\right]\). The column vectors are: \(C_1 = \left[\begin{array}{c} 2 \\ 5 \end{array}\right], \; C_2 = \left[\begin{array}{c} 10 \\ -1 \end{array}\right], \; \text{and} \; C_3 = \left[\begin{array}{c} 6 \\ 3 \end{array}\right]\).

Step 1: Row Vectors

From matrix A, there are 2 row vectors which are the horizontal components. To extract the row vectors from the given matrix, place each row in a separate row vector: \[R_1=\left[\begin{array}{ccc} 2 & 10 & 6 \end{array}\right], \quad R_2=\left[\begin{array}{ccc} 5 & -1 & 3 \end{array}\right]\] So, the row vectors are: \[R_1 = \left[\begin{array}{ccc} 2 & 10 & 6 \end{array}\right] \quad \text{and} \quad R_2 = \left[\begin{array}{ccc} 5 & -1 & 3 \end{array}\right]\]

Step 2: Column Vectors

From matrix A, there are 3 column vectors which are the vertical components. To extract the column vectors from the given matrix, place each column in a separate column vector: \[C_1=\left[\begin{array}{c} 2 \\ 5 \end{array}\right], \quad C_2=\left[\begin{array}{c} 10 \\ -1 \end{array}\right], \quad C_3=\left[\begin{array}{c} 6 \\ 3 \end{array}\right]\] So, the column vectors are: \[C_1 = \left[\begin{array}{c} 2 \\ 5 \end{array}\right], \quad C_2 = \left[\begin{array}{c} 10 \\ -1 \end{array}\right], \quad \text{and} \quad C_3 = \left[\begin{array}{c} 6 \\ 3 \end{array}\right]\]

Key Concepts

Row Vectors

Row vectors are an essential component of matrices, representing the horizontal arrangement of numbers. In a matrix, row vectors run left to right. Each row in the matrix can be considered a row vector.
This is important because understanding rows helps in various matrix operations, such as addition or multiplication. For example, consider the matrix \( A \):\[ A = \begin{bmatrix} 2 & 10 & 6 \ 5 & -1 & 3 \end{bmatrix} \]This matrix consists of two rows:

• Row 1: \( R_1 = \begin{bmatrix} 2 & 10 & 6 \end{bmatrix} \)
• Row 2: \( R_2 = \begin{bmatrix} 5 & -1 & 3 \end{bmatrix} \)

A row vector is essentially a single-row matrix. It's useful for representing data that naturally aligns in a linear fashion, and they play a significant role in row reduction operations and finding solutions to linear equations.

Column Vectors

Column vectors represent the vertical components of a matrix and are a key aspect in its structure. Every column in a matrix functions as a column vector, arranged from top to bottom.
These are crucial when applying operations like the dot product or when solving systems of linear equations.Take the matrix \( A \) as an example:\[ A = \begin{bmatrix} 2 & 10 & 6 \ 5 & -1 & 3 \end{bmatrix} \]This matrix has three columns:

• Column 1: \( C_1 = \begin{bmatrix} 2 \ 5 \end{bmatrix} \)
• Column 2: \( C_2 = \begin{bmatrix} 10 \ -1 \end{bmatrix} \)
• Column 3: \( C_3 = \begin{bmatrix} 6 \ 3 \end{bmatrix} \)

Column vectors are especially significant when transforming data or equations in fields such as physics and engineering. They are fundamental elements in matrix-vector multiplication, often defining directional properties in space.

Matrix Notation

Matrix notation is a systematic way of representing numerical data in rows and columns. This structure has vast applications in various fields, including computer science, physics, and statistics. Each matrix is made up of elements arranged in a grid-like manner, defined by the number of rows and columns.
In matrix notation, a matrix is written as \( A = [a_{ij}] \), where \( a_{ij} \) represents the elements, with \(i\) being the row index and \(j\) the column index.Example: A 2x3 matrix like \( A \) is defined with 2 rows and 3 columns:\[ A = \begin{bmatrix} 2 & 10 & 6 \ 5 & -1 & 3 \end{bmatrix} \]

• "2x3" indicates 2 rows and 3 columns.
• Each element is identified by its position, such as \( a_{11} = 2 \) (1st row, 1st column) and \( a_{23} = 3 \) (2nd row, 3rd column).

Matrices are powerful tools in computation, used to solve systems of linear equations, perform transformations and organize data sets efficiently. Understanding the notation is crucial for reading, writing, and interpreting matrix operations effectively.