Linear transformations

Which of the following vector functions are linear transformations?

Let $A$ be an $m\times n$-matrix. Show the vector function $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ defined by $T(\mathbf{v})=A\mathbf{v}$ is a linear transformation.

Let $\mathbf{u}\in\mathbb{R}^{n}$ be a fixed vector. Show that the function $T$ defined by $T(\mathbf{v})=\mathbf{v}-\operatorname{proj}_{\mathbf{u}}(\mathbf{v})$ is a linear transformation.

Let $\mathbf{u}\in\mathbb{R}^{n}$ be a fixed non-zero vector. The function $T$ defined by $T(\mathbf{v})=\mathbf{u}+\mathbf{v}$ has the effect of translating all vectors by adding $\mathbf{u}$. Show this is not a linear transformation.