Problem 36 - Differential Equations and Linear Algebra

Question

If \(\mathbf{v}_{1}=(1,-1)\) and \(\mathbf{v}_{2}=(2,1)\) are eigenvectors of the matrix \(A\) corresponding to the eigenvalues \(\lambda_{1}=2, \lambda_{2}=-3,\) respectively, find \(A\left(3 \mathbf{v}_{1}-\mathbf{v}_{2}\right)\).

Step-by-Step Solution

Answer

The result of the given expression \(A\left(3\mathbf{v}_1 - \mathbf{v}_2\right)\) is the vector \(\boxed{(12, -3)}\).

Step 1: Recall the Eigenvector-Eigenvalue Relation

To better understand how to use the given information, remember that when \(\mathbf{v}\) is an eigenvector of matrix A, corresponding to the eigenvalue \(\lambda\), it means the following relationship is true: \[ A\mathbf{v} = \lambda\mathbf{v} .\] We are provided with the eigenvectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\), and their corresponding eigenvalues \(\lambda_1=2\) and \(\lambda_2=-3\), so we know that: \[ A\mathbf{v}_1 = 2\mathbf{v}_1 ,\] and \[ A\mathbf{v}_2 = -3\mathbf{v}_2 .\]

Step 2: Substitute the Relationship into the Given Expression

We need to find \(A\left(3\mathbf{v}_1 -\mathbf{v}_2\right)\). To accomplish this, we substitute the relationships from Step 1 into this expression: \[ A\left(3\mathbf{v}_1 - \mathbf{v}_2\right) = 3A\mathbf{v}_1 - A\mathbf{v}_2 .\]

Step 3: Use the Eigenvector-Eigenvalue Relation

Now, apply the relationships derived in Step 1: \[ 3A\mathbf{v}_1 - A\mathbf{v}_2 = 3(2\mathbf{v}_1) - (-3\mathbf{v}_2) .\]

Step 4: Simplify the Expression

Simplify the expression we got in Step 3: \[ 3(2\mathbf{v}_1) - (-3\mathbf{v}_2) = 6\mathbf{v}_1 + 3\mathbf{v}_2 .\]

Step 5: Replace the Eigenvectors with their Given Values

Finally, substitute the given eigenvectors \(\mathbf{v}_1 = (1, -1)\) and \(\mathbf{v}_2 = (2, 1)\) into the expression: \[ 6\mathbf{v}_1 + 3\mathbf{v}_2 = 6(1,-1) + 3(2,1) = (6,-6) + (6,3) .\]

Step 6: Perform the Vector Addition

Add the vectors obtained in Step 5: \[ (6, -6) + (6, 3) = (6+6, -6+3) = (12, -3) .\] The result of the given expression \(A\left(3\mathbf{v}_1 - \mathbf{v}_2\right)\) is the vector \(\boxed{(12, -3)}\).

Key Concepts

Matrix Operations

Matrix operations involve performing various calculations involving matrices, such as addition, subtraction, and multiplication. This is essential for determining characteristics of linear systems like eigenvectors and eigenvalues.

• Matrix Multiplication: When a matrix multiplies a vector or another matrix, it transforms or alters it. This operation is a rule-governed sequence involving the rows of the first matrix and columns of the second.
• Eigenoperations: In the context of eigenvectors and eigenvalues, matrix operations reveal the fundamental properties of the transformation a matrix provides. For example, the relationship \[ A\mathbf{v} = \lambda\mathbf{v} \] describes how multiplication by the matrix \( A \) scales the vector \( \mathbf{v} \) by a factor of \( \lambda \).

Matrix operations help us unlock the ability to predict transformations in terms of both magnitude and direction, which is crucial for higher-level mathematics and physics applications.

Vector Addition

Vector addition is a fundamental operation in linear algebra, appropriately used when dealing with linear combinations of eigenvectors.

• Adding Vectors: Vectors are added element-wise, meaning you add the corresponding components of each vector together. Given two vectors \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), their sum is \( (u_1 + v_1, u_2 + v_2) \).
• Application to Eigenvectors: In our solution, we used vector addition to combine the scaled eigenvectors. After calculating the scaled versions of \( \mathbf{v}_1 = (1,-1) \) and \( \mathbf{v}_2 = (2,1) \), we perform vector addition to find \( (12, -3) \).

This operation is central to many practical applications, including physics and computer graphics, where combining multiple directional vectors enables complex calculations of movement and positioning.

Linear Transformations

Linear transformations describe the mapping of vectors and matrices that maintain the operations of addition and scalar multiplication. They are linked closely with matrices, which can be viewed as tools for performing these transformations.

• Understanding Transformations: A transformation is linear if it satisfies the following for vectors \( \mathbf{u} \) and \( \mathbf{v} \), and scalar \( c \): \[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \] and \[ T(c\mathbf{u}) = cT(\mathbf{u}) \]
• Role of Eigenvectors: Eigenvectors and eigenvalues simplify the understanding of these transformations by showing how a transformation can stretch or compress space along certain directions (eigenvectors) uniformly (eigenvalues).

This concept of linear transformations is fundamental in understanding how and why different phenomena occur, especially in fields like machine learning and quantum mechanics, where these transformations underpin algorithms and theories.