Dimension statements for spanned subspaces

Let $\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3,\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ be vectors in $\mathbb{R}^4$. Let $U$ be the span of $\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\}$ and let $V$ be the span of $\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$. Consider the following statements: (I) If the dimension of $U \cap V$ is 2 and the dimension of $U$ is 3, then $\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$ is linearly dependent. (II) If $U + V = \{\mathbf{u} + \mathbf{v} : \mathbf{u} \in U, \mathbf{v} \in V\} = \mathbb{R}^4$, then either $\{\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3\}$ is linearly independent or $\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\}$ is linearly independent. Which of the above statements is/are true?