Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 13, 2023

Which linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ is defined by $T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$ ?

STEP 1

Assumptions
1. A linear transformation $T$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ should take a 3-dimensional vector as input and produce a 2-dimensional vector as output.
2. A transformation is linear if it satisfies two properties for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ and any scalar $c$ : - Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ - Homogeneity: $T(c\mathbf{u}) = cT(\mathbf{u})$
3. The transformation should not include any operations that violate linearity, such as squaring a variable, multiplying variables together, or using variables not present in the input vector.

STEP 2

Examine the first transformation:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{r} x_{1}^{2}+x_{2} \\ x_{1}-x_{2} \\ x_{3} \end{array}\right]$
This transformation cannot be linear because it includes a squared term $x_{1}^{2}$ , which violates the homogeneity property, and it produces a 3-dimensional vector instead of a 2-dimensional one.

STEP 3

Examine the second transformation:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$
This transformation takes a 3-dimensional vector and produces a 2-dimensional vector by simply taking the first two components of the input vector. This operation satisfies both additivity and homogeneity, so it is a candidate for a linear transformation.

STEP 4

Examine the third transformation:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{r} x_{1}+x_{2} \\ x_{3}-x_{1} \\ x_{2} \end{array}\right]$
This transformation cannot be linear because it takes a 2-dimensional vector as input and produces a 3-dimensional vector as output, which does not match the required dimensions for a transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ . Additionally, it includes a variable $x_{3}$ that is not present in the input vector.

STEP 5

Examine the fourth transformation:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{r} x_{1} x_{2} \\ x_{1}+x_{2}+x_{3} \end{array}\right]$
This transformation cannot be linear because it includes a product of variables $x_{1} x_{2}$ , which violates the homogeneity property.

STEP 6

Based on the examination of all four transformations, the only transformation that satisfies the properties of a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$ is the second one:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$
This is because it takes a 3-dimensional vector as input, produces a 2-dimensional vector as output, and satisfies both additivity and homogeneity properties of a linear transformation.
The correct linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ is:
$T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]$

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