Column vectors are written as rows, such as 𝐱=(x1, x2), and T(𝐱) is written as T(x1, x2). Show t

step 1 In this video, we're going to be working with the transformation T that we see defined here in r

Column vectors are written as rows, such as 𝐱=(x1, x2), and...

Column vectors are written as rows, such as $\mathbf{x}=\left(x_{1}, x_{2}\right),$ and $T(\mathbf{x})$ is written as $T\left(x_{1}, x_{2}\right)$. Show that the transformation $T$ defined by $T\left(x_{1}, x_{2}\right)=$ $\left(2 x_{1}-3 x_{2}, x_{1}+4,5 x_{2}\right)$ is not linear.

Column vectors are written as rows, such as $\mathbf{x}=\left(x_{1}, x_{2}\right),$ and $T(\mathbf{x})$ is written as $T\left(x_{1}, x_{2}\right)$.
Show that the transformation $T$ defined by $T\left(x_{1}, x_{2}\right)=$ $\left(2 x_{1}-3 x_{2}, x_{1}+4,5 x_{2}\right)$ is not linear.

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any two vectors u and v and any scalar c, the transformation must satisfy T(u + v) = T(u) + T(v) and T(c · u) = c · T(u).

Properties of Linearity (Additivity and Homogeneity)

These are the two essential properties that a transformation must adhere to in order to be considered linear. Additivity requires that the transformation of a sum of vectors equals the sum of their individual transformations, while homogeneity requires that scaling a vector before applying the transformation yields the same result as applying the transformation first and then scaling.

Zero Vector Mapping

A necessary condition for linearity is that the transformation maps the zero vector in the domain to the zero vector in the codomain (i.e., T(0)=0). This property directly follows from the additivity and homogeneity requirements and serves as a quick test for non-linearity.

Translation (Offset) in Transformations

The presence of a constant term or fixed vector addition (an offset) in the definition of a transformation often indicates that the transformation is affine rather than linear. This additional constant component violates the necessary condition that the zero vector must map to the zero vector, hence breaking linearity.

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