Which of the following functions F: ℝ^3 →ℝ are linear? (a) F(x, y, z)=x, (b) F(x, y, z)=y-2, (c)

step 1 To find the partial derivative with respect to S, we treat Y and Z as constant. So, the partial

Which of the following functions F: ℝ^3 →ℝ are linear? (a)...

Which of the following functions $F: \mathbb{R}^3 \rightarrow \mathbb{R}$ are linear? (a) $F(x, y, z)=x$,
(b) $F(x, y, z)=y-2$, (c) $F(x, y, z)=x+y+3$, (d) $F(x, y, z)=x-y-z$,
(e) $F(x, y, z)=x y z$,
(f) $F(x, y, z)=x^2-y^2+z^2$,
(g) $F(x, y, z)=e^{x-y+z}$.

Key Concepts

Additivity

Additivity is one of the key properties that a function must satisfy to be considered linear. It requires that the function applied to the sum of two vectors yields the same result as the sum of the function applied to each vector individually. This property ensures that the structure of vector addition in the domain is maintained in the codomain.

Homogeneity

Homogeneity, also known as scalar multiplication preservation, is another essential property of a linear function. It requires that scaling an input vector by a scalar results in the output being scaled by the same scalar, i.e., F(c * v) = c * F(v) for any scalar c. This property ensures that the function responds predictably to changes in the magnitude of the input vectors.

Linear Transformation

A linear transformation, or linear function, is a mapping between vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any vectors u and v and any scalar c, the transformation F must satisfy F(u + v) = F(u) + F(v) and F(cu) = cF(u). In these types of functions, the output is a linear combination of the inputs with no additional constant terms or non-linear operations.

Affine Functions

Affine functions are functions that are similar to linear functions but include an additional constant term. They have the form F(x) = L(x) + b, where L is a linear function and b is a constant vector or scalar. Although affine functions preserve the structure of the vector space in many ways, they do not meet the strict definition of linearity because they do not necessarily send the zero vector to zero.

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