Problem 16 - Differential Equations and Linear Algebra

Question

Determine whether the given set of vectors is linearly independent in \(M_{2}(\mathbb{R})\). $$A_{1}=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right], A_{2}=\left[\begin{array}{rr} 2 & -1 \\ 0 & 1 \end{array}\right], A_{3}=\left[\begin{array}{ll} 3 & 6 \\ 0 & 4 \end{array}\right]$$.

Step-by-Step Solution

Answer

The given set of matrices \(A_1, A_2, A_3\) in \(M_{2}(\mathbb{R})\) is linearly dependent. This is concluded after performing Gaussian elimination on the combined matrix, which resulted in only two pivots, less than the number of matrices.

Step 1: Write the augmented matrix

First, combine the matrices into a single matrix like this: \[ \begin{bmatrix} 1 & 1 & 2 & -1 & 3 & 6 \\ 0 & 1 & 0 & 1 & 0 & 4 \end{bmatrix} \]

Step 2: Perform Gaussian elimination

Now, perform Gaussian elimination to find the row echelon form of the matrix. We can start by using the first row to eliminate the first elements in the second and third columns: \[ \begin{bmatrix} 1 & 1 & 0 & -3 & 0 & -10 \\ 0 & 1 & 0 & 1 & 0 & 4 \end{bmatrix} \]

Step 3: Identify the pivots

In the row echelon form of the matrix, there are two pivots: one in the first row and one in the second row.

Step 4: Conclude about linear independence

Since there are only two pivots in the row echelon form, which is less than the number of matrices (three), we conclude that the given set of matrices is linearly dependent.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It is also instrumental in transforming matrices into simpler forms, like the Row Echelon Form (REF). This process makes it easier to understand and solve systems or evaluate matrix properties, such as linear independence.

In Gaussian Elimination, we systematically perform row operations to achieve a triangular form. Here are some vital steps involved:

• Identify a pivot element that is non-zero. It is typically the first non-zero number in the first column.
• Swap rows if necessary to move the pivot element to the top row.
• Eliminate other entries in the pivot's column, by adding or subtracting multiples of the pivot row to other rows.
• Repeat the process for subsequent rows, working down the matrix.

Mastering Gaussian Elimination helps in reducing any matrix into a more manageable form, such as Row Echelon Form, which can reveal properties like rank and linear independence.

Matrix Theory

Matrix Theory is a branch of mathematics focusing on the study and application of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions, organized in rows and columns. They provide a powerful tool for various fields including physics, computer science, and economics.

In the context of linear independence, matrices represent systems where we want to determine if the vectors represented by the matrix's columns are independent. Key concepts in matrix theory include:

• Dimension: Defines the matrix size as rows by columns (e.g., a 2x3 matrix).
• Determinants: A scalar value that can indicate certain properties about the matrix, such as invertibility.
• Rank: The number of independent rows or columns, which helps determine the matrix's span.

Understanding these concepts is essential for solving problems involving linear equations, such as the one in the original exercise, where the goal is to assess the linear independence of a set of matrices.

Row Echelon Form

Row Echelon Form (REF) is a matrix that has been manipulated through Gaussian Elimination. It is characterized by a staircase-like pattern formed by leading elements known as pivots. A matrix is in row echelon form if:

• All nonzero rows are above any rows of all zeroes.
• Every leading entry of a row is in a column to the right of the leading entry of the row above it.
• The leading entry in any nonzero row is 1 and is called a pivot.

These features help to simplify calculations and conclusions regarding linear independence or dependence.

In the context of the exercise provided, after converting the system of matrices into row echelon form, the number of pivots directly corresponds to the number of linearly independent vectors. If there are fewer pivots than vectors, as in this exercise, the vectors are linearly dependent. Row Echelon Form simplifies identifying these pivots, thereby streamlining conclusions about linear systems or sets.