Which of the following functions F: ℝ^2 →ℝ^2 are linear? (a) F( x y )=( x-y

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Which of the following functions F: ℝ^2 →ℝ^2 are linear? (a)...

Which of the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are linear? (a) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x-y \\ x+y\end{array}\right)$, (b) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+y+1 \\ x-y-1\end{array}\right)$, (c) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x y \\ x-y\end{array}\right)$, (d) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}3 y \\ 2 x\end{array}\right)$, (e) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x^2+y^2 \\ x^2-y^2\end{array}\right)$, (f) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}y-3 x \\ x\end{array}\right)$.

Which of the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are linear?
(a) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x-y \\ x+y\end{array}\right)$,
(b) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+y+1 \\ x-y-1\end{array}\right)$,
(c) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x y \\ x-y\end{array}\right)$,
(d) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}3 y \\ 2 x\end{array}\right)$,
(e) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x^2+y^2 \\ x^2-y^2\end{array}\right)$,
(f) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}y-3 x \\ x\end{array}\right)$.

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Key Concepts

Matrix Representation of Linear Maps

For finite-dimensional vector spaces, any linear transformation can be represented by a matrix. This representation allows for a concrete computation of the transformation by multiplying the matrix with coordinate vectors. This central concept simplifies the study and verification of linear properties using linear algebra techniques.

Additivity and Homogeneity

The key properties used to test linearity are additivity (F(u + v) = F(u) + F(v)) and homogeneity (F(cu) = cF(u)). These properties are necessary and sufficient conditions for a function between vector spaces to be considered linear. They ensure that the function respects the structure of the vector space, allowing for consistent manipulation and analysis.

Linear Transformation

A linear transformation between vector spaces is a function that preserves the operations of vector addition and scalar multiplication. This means for all vectors u, v and any scalar c, the transformation satisfies F(u + v) = F(u) + F(v) and F(cu) = cF(u). Linear transformations are fundamental in understanding the structure and properties of vector spaces.

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