(Solved): Are the following statements true or false? 1. For any scalar \( c \), and ve ...

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Are the following statements true or false? 1. For any scalar \( c \), and vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^{n} \), we have \( \mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \). 2. For any vector \( \mathbf{v} \in \mathbb{R}^{n} \), we have \( \mathbf{v} \cdot \mathbf{v}=\|\mathbf{v}\| \). 3. For all vectors \( \mathbf{u}, \mathbf{v} \in \mathbb{R}^{n} \), we have \( \mathbf{u} \cdot \mathbf{v}=-\mathbf{v} \cdot \mathbf{u} \). 4. For a square matrix \( A \), vectors in the column space of \( A \) are orthogonal to vectors in the nullspace of \( A \). 5. If \( \mathbf{x} \) is not in a subspace \( W \), then \( \mathbf{x}-\operatorname{proj}_{W}(\mathbf{x}) \) is zero.