Problem 1 - Differential Equations and Linear Algebra

Question

Consider \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}\) defined by \(T(\mathbf{x})=A \mathbf{x}\) where \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 4 \\ 4 & 8 \\ 8 & 16\end{array}\right] .\) For each \(\mathbf{x}\) below, find \(T(\mathbf{x})\) and thereby determine whether \(\mathbf{x}\) is in \(\operatorname{Ker}(T).\) (a) \(\mathbf{x}=(-10,5).\) (b) \(\mathbf{x}=(1,-1).\) (c) \(\mathbf{x}=(2,-1).\)

Step-by-Step Solution

Answer

For the given vectors: (a) \(\mathbf{x_1}=(-10,5)\) belongs to the kernel of T, as \(T(\mathbf{x_1}) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). (b) \(\mathbf{x_2}=(1,-1)\) does not belong to the kernel of T, since \(T(\mathbf{x_2}) \neq \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\). (c) \(\mathbf{x_3}=(2,-1)\) belongs to the kernel of T, as \(T(\mathbf{x_3}) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\).

Step 1: Setup given matrix and vectors

The given matrix A and vectors from (a), (b), (c) are as follows: Matrix A: \[ \begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 4 & 8 \\ 8 & 16 \end{bmatrix} \] Vectors x: (a) \(\mathbf{x_1}=(-10,5)\) (b) \(\mathbf{x_2}=(1,-1)\) (c) \(\mathbf{x_3}=(2,-1)\)

Step 2: Calculate T(x) for x1, x2, and x3

We will now multiply matrix A with each vector x to find T(x). (a) For \(\mathbf{x_1}=(-10,5)\) \(T(\mathbf{x_1}) = A\mathbf{x_1}=\left[\begin{array}{ll}1 & 2 \\\ 2 & 4 \\\ 4 & 8 \\\ 8 & 16\end{array}\right]\begin{bmatrix} -10 \\ 5 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \) (b) For \(\mathbf{x_2}=(1,-1)\) \(T(\mathbf{x_2}) = A\mathbf{x_2}=\left[\begin{array}{ll}1 & 2 \\\ 2 & 4 \\\ 4 & 8 \\\ 8 & 16\end{array}\right]\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ -2 \\ -4 \\ -8 \end{bmatrix} \) (c) For \(\mathbf{x_3}=(2,-1)\) \(T(\mathbf{x_3}) = A\mathbf{x_3}=\left[\begin{array}{ll}1 & 2 \\\ 2 & 4 \\\ 4 & 8 \\\ 8 & 16\end{array}\right]\begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)

Step 3: Check whether given x belongs to the kernel of T

If T(x) is the zero vector, then x belongs to the kernel of T. (a) For \(\mathbf{x_1}=(-10,5)\), since \(T(\mathbf{x_1}) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\), it belongs to the kernel of T. (b) For \(\mathbf{x_2}=(1,-1)\), since \(T(\mathbf{x_2}) \neq \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\), it does not belong to the kernel of T. (c) For \(\mathbf{x_3}=(2,-1)\), since \(T(\mathbf{x_3}) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}\), it belongs to the kernel of T.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra and is essential when dealing with linear transformations. When you multiply a matrix by a vector, you are essentially applying a linear transformation to that vector. This operation adjusts the vector to a new position in a different vector space.

Here's how the multiplication process works:

• Each element of the resulting vector is a sum of products. You take the corresponding elements from the row of the matrix and column of the vector and multiply them.
• The number of columns in the matrix must match the number of rows in the vector.

For the transformation defined in the exercise, matrix A is multiplied by vectors \( \mathbf{x_1}, \mathbf{x_2}, \text{and} \mathbf{x_3} \) to create new vectors. Notice how when matrix A, which is \([4 imes 2]\), multiplies vectors, it transforms a 2-dimensional vector into a 4-dimensional vector.

Matrix multiplication is essential for understanding how transformations map vectors to different spaces, especially when analyzing their effects on vector spaces like kernels.

Kernel of a Transformation

The kernel of a transformation is an important concept in linear algebra, especially when evaluating whether certain vectors get mapped to the zero vector after a linear transformation.

The kernel, or null space, is the set of all input vectors that the transformation maps to the zero vector. Determining if a vector is in the kernel involves the following:

• Apply the transformation to the vector.
• If the resulting vector is the zero vector, the original vector is part of the kernel.

In the given exercise, vectors \( \mathbf{x_1} \) and \( \mathbf{x_3} \) mapped to the zero vector when matrix A was applied. Hence, they are part of the kernel of the transformation T. Understanding the kernel is crucial as it provides insights into the vector space where the transformation acts trivially, i.e., the transformation has no effect.

Knowing the kernel is useful for solving linear equations and identifying dependencies within transformations. It is a foundational aspect when exploring vector space properties.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, vector spaces, and linear transformations. It provides the academic framework for understanding how systems of linear equations can be represented and solved using matrices.

One of the key highlights of linear algebra is its role in describing linear transformations using matrices and vectors:

• Vectors are entities with magnitude and direction. They can represent points, directions, or quantities in space.
• Matrices serve as linear transformations. They can rotate, stretch, compress, or otherwise alter vectors.
• The concepts of bases, dimensions, row space, column space, and kernel are pivotal in understanding vector spaces.

Linear algebra relies on concepts like matrix multiplication and transformations to explore the structure of vector spaces. It finds applications in many fields, including computer graphics, engineering, and data science.

Through the lens of linear algebra, the problem is approached by evaluating how the transformation T maps vectors from a 2-dimensional space into a 4-dimensional space using the given matrix A. Understanding these mappings is fundamental to linear algebra as it reveals the behavior and properties of transformations in higher dimensions.