Question

Let \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) be a linear transformation. a. Find \(T\left[\begin{array}{l}8 \\ 3 \\ 7\end{array}\right]\) if \(T\left[\begin{array}{r}1 \\ 0 \\\ -1\end{array}\right]=\left[\begin{array}{l}2 \\ 3\end{array}\right]\) and \(T\left[\begin{array}{l}2 \\ 1 \\\ 3\end{array}\right]=\left[\begin{array}{r}-1 \\ 0\end{array}\right]\). b. Find \(T\left[\begin{array}{r}5 \\ 6 \\ -13\end{array}\right]\) if \(T\left[\begin{array}{r}3 \\ 2 \\\ -1\end{array}\right]=\left[\begin{array}{l}3 \\ 5\end{array}\right]\) and \(T\left[\begin{array}{l}2 \\ 0 \\\ 5\end{array}\right]=\left[\begin{array}{r}-1 \\ 2\end{array}\right]\)

Short answer

Part (a): \( \begin{bmatrix} 1 \\ 6 \end{bmatrix} \); Part (b): \( \begin{bmatrix} 11 \\ 11 \end{bmatrix} \)."

Step-by-step solution

Express Vectors as a Combination of Given Vectors (Part a)

To find the image of \( \begin{bmatrix} 8 \ 3 \ 7 \end{bmatrix} \) under \( T \), we will express this vector as a linear combination of the known vectors. Let \( \begin{bmatrix} 8 \ 3 \ 7 \end{bmatrix} = a \begin{bmatrix} 1 \ 0 \ -1 \end{bmatrix} + b \begin{bmatrix} 2 \ 1 \ 3 \end{bmatrix} \). This yields equations: 1. \( a + 2b = 8 \)2. \( b = 3 \)3. \( -a + 3b = 7 \). Solve from the second equation: \( b = 3 \). Substitute in the first equation: \( a + 2(3) = 8 \) leading to \( a = 2 \). Check the third equation: \( -2 + 3(3) = 7 \) confirms this solution.

Apply Linear Transformation to the Combination (Part a)

Now use the found coefficients to compute \( T \left[ \begin{bmatrix} 8 \ 3 \ 7 \end{bmatrix} \right] = aT \left[ \begin{bmatrix} 1 \ 0 \ -1 \end{bmatrix} \right] + bT \left[ \begin{bmatrix} 2 \ 1 \ 3 \end{bmatrix} \right] \).Substitute \( a = 2 \) and \( b = 3 \):\[\begin{align*}T \left[ \begin{bmatrix} 8 \ 3 \ 7 \end{bmatrix} \right] &= 2 \begin{bmatrix} 2 \ 3 \end{bmatrix} + 3 \begin{bmatrix} -1 \ 0 \end{bmatrix} \&= \begin{bmatrix} 4 \ 6 \end{bmatrix} + \begin{bmatrix} -3 \ 0 \end{bmatrix} \&= \begin{bmatrix} 1 \ 6 \end{bmatrix}.\end{align*}\]

Express Vectors as a Combination of Given Vectors (Part b)

Similarly, express \( \begin{bmatrix} 5 \ 6 \ -13 \end{bmatrix} \) as \( a \begin{bmatrix} 3 \ 2 \ -1 \end{bmatrix} + b \begin{bmatrix} 2 \ 0 \ 5 \end{bmatrix} \). Solve the system: 1. \( 3a + 2b = 5 \)2. \( 2a = 6 \)3. \( -a + 5b = -13 \).From the second equation: \( a = 3 \). Substitute in the first: \( 3(3) + 2b = 5 \) leading to \( 2b = -4 \) thus \( b = -2 \). Verify the third equation: \( -3 + 5(-2) = -13 \) verifies this solution.

Apply Linear Transformation to the Combination (Part b)

Compute \( T \left[ \begin{bmatrix} 5 \ 6 \ -13 \end{bmatrix} \right] = aT \left[ \begin{bmatrix} 3 \ 2 \ -1 \end{bmatrix} \right] + bT \left[ \begin{bmatrix} 2 \ 0 \ 5 \end{bmatrix} \right] \).Substitute \( a = 3 \) and \( b = -2 \):\[\begin{align*}T \left[ \begin{bmatrix} 5 \ 6 \ -13 \end{bmatrix} \right] &= 3 \begin{bmatrix} 3 \ 5 \end{bmatrix} + (-2) \begin{bmatrix} -1 \ 2 \end{bmatrix} \&= \begin{bmatrix} 9 \ 15 \end{bmatrix} + \begin{bmatrix} 2 \ -4 \end{bmatrix} \&= \begin{bmatrix} 11 \ 11 \end{bmatrix}.\end{align*}\]

Conclude the Results

The images of the given vectors under transformation \( T \) are:- Part (a): \( T \left[ \begin{bmatrix} 8 \ 3 \ 7 \end{bmatrix} \right] = \begin{bmatrix} 1 \ 6 \end{bmatrix} \).- Part (b): \( T \left[ \begin{bmatrix} 5 \ 6 \ -13 \end{bmatrix} \right] = \begin{bmatrix} 11 \ 11 \end{bmatrix} \).

Key concepts

Linear Transformation

In linear algebra, a linear transformation is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. Simply put, if you have two vectors and a scalar (a constant number), a linear transformation will do the following:

• Add vectors together and then transform them, resulting in the same outcome as if you had transformed each vector first and then added them.
• Multiply a vector by a scalar before transforming it, yielding the same result as transforming the vector first and then multiplying the transformed vector by the scalar.

In the given exercise, the transformation function is denoted as \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^2\), which means it takes a 3-dimensional input and maps it to a 2-dimensional output. Understanding how this mapping works is crucial in predicting the output of different vector inputs.

Linear Combination

The concept of linear combination is a way of constructing new vectors by combining given vectors using specific coefficients (which can be any real numbers). You multiply each given vector by a coefficient and add them up. The expression \(a\begin{bmatrix} 1 \ 0 \ -1 \end{bmatrix} + b\begin{bmatrix} 2 \ 1 \ 3 \end{bmatrix}\), as used in the exercise, is an example of a linear combination. Here, \(a\) and \(b\) are coefficients that allow you to express the target vector \([8, 3, 7]\) as a combination of two given vectors.

• First, find coefficients that match the target vector when combined.
• These coefficients are then used to find the expression of the desired vector.

This technique is essential in vector spaces for determining spanning sets, bases, and for transformations like the one in the problem.

Matrix Equations

Matrix equations involve solving for an unknown matrix, often using systems of linear equations. In the exercise, converting the vectors into a system of equations helps find the coefficients \(a\) and \(b\) for the linear combination. For example, the equation \(a + 2b = 8\) is directly derived from the elements of the given vectors.

• Solve the system of equations step-by-step to find all unknowns.
• Substitute them back to verify your solutions.

Matrix equations are fundamental in linear algebra as they allow performing complex operations such as transformation, scaling, rotation, and more in a structured and simplified manner.

Vector Spaces

Vector spaces are mathematical structures that are formed by a collection of vectors. They involve two operations: vector addition and scalar multiplication. In a vector space like \(\mathbb{R}^3\), vectors are defined and manipulated under these operations according to certain rules. Each vector in the space can be a linear combination of a set of basis vectors, which is key in the exercise where transformations from 3D to 2D are dealt with.

• Understand that any vector in a vector space can be expressed as a linear combination of others.
• The vector space provides a backdrop for understanding transformations and mappings from one space to another.

Recognizing the properties and dimensions of vector spaces helps in tasks including solving matrix equations and applying transformations efficiently.