Q1: Is the family $(\mathbf{v})$ consisting of a single vector linearly independent?

A1: Yes, if $ \mathbf{v} \neq 0$.

A2: No, never!

A3: Yes, always!

A4: Yes, if $ \mathbf{v} = 0$.

A5: Yes, if $ \mathbf{v} \in \mathbb{R}^2$.

Q2: Is a family consisting of $(n+1)$-many vectors from $\mathbb{R}^n$ linearly independent?

A1: Yes, if one vector $ \mathbf{v} \neq 0$.

A2: No, never!

A3: Yes, always!

A4: Yes, if one vector $ \mathbf{v} = 0$.

A5: Yes, if $n = 2$.

Q3: Let $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ be a family of vectors in $\mathbb{R}^3$ that are linearly independent. Which claim is not correct?

A1: $ \mathrm{Span}( \mathbf{u}, \mathbf{v}, \mathbf{w} ) = \mathrm{Span}( \mathbf{u}, \mathbf{w} ) $

A2: $ \mathrm{Span}( \mathbf{u}, \mathbf{v}, \mathbf{w} ) = \mathrm{Span}( \mathbf{u}, \mathbf{v}, \mathbf{w} , \mathbf{w} ) $

A3: $ \mathrm{Span}( \mathbf{u}, \mathbf{v}, \mathbf{w} ) = \mathrm{Span}( \mathbf{u}, \mathbf{w} , \mathbf{v} ) $

A4: $ \mathrm{Span}( \mathbf{u}, \mathbf{v}, \mathbf{w} ) = \mathrm{Span}( \mathbf{u}, \mathbf{v} , \mathbf{w} , \mathbf{0} ) $

A5: One needs more information.