Are the following linear functions? Prove your conclusions...

Are the following linear functions? Prove your conclusions by showing that $f(\mathbf{r})$ satisfies both of equations ( $7.1$ ) or that it does not satisfy at least one of them.
$f(\mathbf{r})=\mathbf{A} \cdot \mathbf{r}+3$, where $\mathbf{A}$ is a given vector.

Key Concepts

Homogeneity Property

Homogeneity means that if a vector is scaled by a scalar, then the function's output is scaled by the same scalar. For a function f, this property is expressed as f(c*r) = c*f(r). In the given function, the constant term 3 does not scale with the input, causing the function to violate the homogeneity condition required for linearity.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar, often representing a projection of one vector onto another. When used in a function like f(r) = A · r, it yields a linear mapping if A is fixed. However, the introduction of an additional constant term makes the overall function non-linear, even though the dot product component itself is linear.

Additivity Property

Additivity requires that the function applied to the sum of two vectors equals the sum of the function applied to each vector separately. In the context of the given function f(r) = A · r + 3, the additional constant disrupts this equality because f(u + v) will include an extra 3, while f(u) + f(v) will include two copies of 3.

Linear Function

A linear function (or linear transformation) between vector spaces must satisfy both the additivity and homogeneity properties. This means that for any vectors u and v, and any scalar c, the function must satisfy f(u + v) = f(u) + f(v) and f(c*u) = c*f(u). An important consequence of these properties is that a linear function must map the zero vector to zero, i.e., f(0) = 0.

Affine Function

An affine function is similar to a linear function but includes an additional constant term. While the linear part of an affine function satisfies additivity and homogeneity, the constant term generally causes the function to fail these conditions unless it is zero. Therefore, adding a constant to a linear expression yields an affine transformation rather than a strictly linear function.

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