Explain why the following functions F: ℝ^2 →ℝ^2 are not linear. (a) ( x+2 x+y ), (b) (

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Explain why the following functions F: ℝ^2 →ℝ^2 are not...

Explain why the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are not linear. (a) $\left(\begin{array}{l}x+2 \\ x+y\end{array}\right)$, (b) $\left(\begin{array}{l}x^2 \\ y^2\end{array}\right)$, (c) $\left(\begin{array}{l}|y| \\ |x|\end{array}\right)$, (d) $\left(\begin{array}{c}\sin (x+y) \\ x-y\end{array}\right)$, (e) $\left(\begin{array}{l}x+e^y \\ 2 x+y\end{array}\right)$.

Explain why the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are not linear.
(a) $\left(\begin{array}{l}x+2 \\ x+y\end{array}\right)$,
(b) $\left(\begin{array}{l}x^2 \\ y^2\end{array}\right)$,
(c) $\left(\begin{array}{l}|y| \\ |x|\end{array}\right)$,
(d) $\left(\begin{array}{c}\sin (x+y) \\ x-y\end{array}\right)$,
(e) $\left(\begin{array}{l}x+e^y \\ 2 x+y\end{array}\right)$.

Key Concepts

Nonlinear Operations

Nonlinear operations modify the input in ways that do not adhere to the additivity or homogeneity properties required for linearity. Examples include applying functions like squaring, taking absolute values, or applying trigonometric functions such as sine. These operations change the relationship between input and output so that the transformation does not distribute over addition or scalar multiplication.

Affine Function

An affine function is a function that can be written as a linear function plus a constant vector. While every linear function is affine, not all affine functions are linear because the presence of a nonzero constant term breaks the homogeneity condition required for linearity. Thus, adding a constant term to a linear combination prevents the function from being strictly linear.

Superposition Principle

The superposition principle is a direct consequence of the properties of a linear transformation since it requires that the transformation of a linear combination of inputs is equal to the same linear combination of the transformations. When a function violates this principle—by introducing terms that depend non-linearly on the inputs—it is not linear.

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. In other words, for a function T to be linear, it must satisfy T(u + v) = T(u) + T(v) for any vectors u and v, and T(cu) = cT(u) for any scalar c and vector u. This set of properties ensures that the operation is compatible with the structure of the vector space.

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