###### Exercise5

Determine whether the following statements are true or false and provide a justification for your response. Throughout, we will assume that the matrix $$A$$ has columns $$\vvec_1,\vvec_2,\ldots,\vvec_n\text{;}$$ that is,

\begin{equation*} A = \left[\begin{array}{rrrr} \vvec_1\amp\vvec_2\amp\ldots\amp\vvec_n \end{array}\right]\text{.} \end{equation*}
1. If the equation $$A\xvec = \bvec$$ is consistent, then $$\bvec$$ is in $$\span{\vvec_1,\vvec_2,\ldots,\vvec_n}\text{.}$$

2. The equation $$A\xvec = \vvec_1$$ is always consistent.

3. If $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ $$\vvec_3\text{,}$$ and $$\vvec_4$$ are vectors in $$\real^3\text{,}$$ then their span is $$\real^3\text{.}$$

4. If $$\bvec$$ can be expressed as a linear combination of $$\vvec_1, \vvec_2,\ldots,\vvec_n\text{,}$$ then $$\bvec$$ is in $$\span{\vvec_1,\vvec_2,\ldots,\vvec_n}\text{.}$$

5. If $$A$$ is a $$8032\times 427$$ matrix, then the span of the columns of $$A$$ is a set of vectors in $$\real^{427}\text{.}$$

in-context