Question

In each case, find a linear transformation with the given properties and compute \(T(\mathbf{v})\) $$ \begin{array}{l} \text { a. } T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} ; T(1,2)=(1,0,1) \\ \quad T(-1,0)=(0,1,1) ; \mathbf{v}=(2,1) \\ \text { b. } T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} ; T(2,-1)=(1,-1,1) \\\ \quad T(1,1)=(0,1,0) ; \mathbf{v}=(-1,2) \\ \text { c. } T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{3} ; T\left(x^{2}\right)=x^{3}, T(x+1)=0 \\ \quad T(x-1)=x ; \mathbf{v}=x^{2}+x+1 \\ \text { d. } T: \mathbf{M}_{22} \rightarrow \mathbb{R} ; T\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]=3, T\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=-1, \\ \quad T\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right]=0=T\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right] ; \mathbf{v}=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \end{array} $$

Short answer

The transformed vector \( T(\mathbf{v}) \) is \( \left( \frac{1}{2}, -\frac{3}{2}, -1 \right) \).

Step-by-step solution

Understanding the Linear Transformation

For part a, we need a transformation matrix that maps vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \). We have the conditions:\[ T(1,2) = (1,0,1) \]\[ T(-1,0) = (0,1,1) \]

Set Up the System

Express the transformation as a matrix \( A \) that satisfies:\[ A \begin{bmatrix} 1 \ 2 \end{bmatrix} = \begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix} \]\[ A \begin{bmatrix} -1 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 1 \ 1 \end{bmatrix} \]

Write the Matrix Equation

Form the matrix equation that represents the system:\[ A = \begin{bmatrix} a & b \ c & d \ e & f \end{bmatrix} \]Substitute the conditions to get equations for \( a, b, c, d, e, \) and \( f \).

Solve for the Transformation Matrix

From the equations:1. \( a + 2b = 1 \)2. \( c + 2d = 0 \)3. \( e + 2f = 1 \)4. \( -a = 0 \)5. \( -c = 1 \)6. \( -e = 1 \)Solve for variables:\[ a = 0, \; b = \frac{1}{2}, \; c = -1, \; d = \frac{1}{2}, \; e = -1, \; f = 1 \]

Construct the Matrix A

The transformation matrix is:\[ A = \begin{bmatrix} 0 & \frac{1}{2} \ -1 & \frac{1}{2} \ -1 & 1 \end{bmatrix} \]

Compute \( T(\mathbf{v}) \)

For \( \mathbf{v} = (2,1) \), compute:\[ T(\mathbf{v}) = A \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 0 & \frac{1}{2} \ -1 & \frac{1}{2} \ -1 & 1 \end{bmatrix} \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 0 \cdot 2 + \frac{1}{2} \cdot 1 \ -1 \cdot 2 + \frac{1}{2} \cdot 1 \ -1 \cdot 2 + 1 \cdot 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \ -\frac{3}{2} \ -1 \end{bmatrix} \]

Key concepts

Matrix Representation

Matrix representation is a powerful technique used in the field of linear algebra to represent linear transformations. A matrix can be thought of as a grid of numbers, where each number plays a role in transforming input vectors into output vectors. This is incredibly useful for computations in higher dimensions.
In our exercise, the transformation from \( \mathbb{R}^2 \to \mathbb{R}^3 \) is represented by a matrix. We constructed a matrix \( A \) such that when it's multiplied by input vectors in \( \mathbb{R}^2 \) space, it yields output vectors in \( \mathbb{R}^3 \) space. This transformation matrix, \( A \), has a specific configuration determined by given conditions, such as how certain vectors map to others. In the exercise, the entries of this matrix were calculated based on the mapping conditions \( T(1,2)= (1,0,1) \) and \( T(-1,0)=(0,1,1) \).
Understanding the structure and formation of such matrices allows one to realize and apply transformations efficiently and accurately.

Vector Transformation

Vector transformation involves altering a vector's state by a given linear transformation. This operation is key in understanding how objects and data can be manipulated in vector spaces. In our exercise, the task is to evaluate the transformed vector \( T(\mathbf{v}) \) given specific conditions.
To do this, you first identify the transformation matrix \( A \), which directly affects the vector \( \mathbf{v} \). Specifically, since \( T(\mathbf{v}) = A \mathbf{v} \), each element in \( \mathbf{v} \) is transformed according to operations dictated by the rows of matrix \( A \).
In the exercise, we computed the transformation for \( \mathbf{v} = (2,1) \) using the transformation matrix \( A \). Each component of \( \mathbf{v} \) underwent changes to produce a new vector based on the polynomial equations derived from matrix multiplication, resulting in \( T(\mathbf{v}) \) as specified in the exercise.

System of Equations

When discussing linear transformations, the formation of a system of equations plays a vital role. It allows us to determine the unknown elements of the transformation matrix. In the given exercise, the conditions specified by how certain vectors must transform define a system of linear equations.
For instance, with the conditions \( T(1,2)=(1,0,1) \) and \( T(-1,0)=(0,1,1) \), we developed a system of equations to solve for the components \( a, b, c, d, e, \) and \( f \) of a 3x2 matrix \( A \).

• These equations are derived from the requirement that multiplying the matrix by given vectors equals specified outcome vectors.
• Finally, solving the system gives the values of the matrix components, enabling the transformation to act correctly.

This process is fundamental in establishing the logic behind transforming input vectors to desired output vectors.

Transformation Matrix

The transformation matrix is the cornerstone in the study of linear transformations, acting as a blueprint for how vectors are manipulated as they move from one space to another. This concept brings about the transformation process itself.
In a typical setup like our exercise, the transformation matrix \( A \) was pivotal in mapping vectors from \( \mathbb{R}^2 \) to \( \mathbb{R}^3 \).

• Each column of \( A \) can be seen as the transformed version of the standard basis vectors of the domain.
• This ensures that any vector from the original space can be transformed predictably and consistently.

The matrix you construct doesn’t only work for a single vector; it holds the code for all vectors in that space, making it an efficient and widely-used tool in various applications, from computer graphics to data analysis.