###### Exercise7

Provide a justification for your response to the following questions.

1. Suppose that we have vectors in $$\real^8\text{,}$$ $$\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{,}$$ whose span is $$\real^8\text{.}$$ Can every vector $$\bvec$$ in $$\real^8$$ be written as a linear combination of $$\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{?}$$

2. Suppose that we have vectors in $$\real^8\text{,}$$ $$\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{,}$$ whose span is $$\real^8\text{.}$$ Can every vector $$\bvec$$ in $$\real^8$$ be written uniquely as a linear combination of $$\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{?}$$

3. Do the vectors

\begin{equation*} \evec_1=\threevec{1}{0}{0}, \evec_2=\threevec{0}{1}{0}, \evec_3=\threevec{0}{0}{1} \end{equation*}

span $$\real^3\text{?}$$

4. Suppose that $$\vvec_1,\vvec_2,\ldots,\vvec_n$$ span $$\real^{438}\text{.}$$ What can you guarantee about the value of $$n\text{?}$$

5. Can 17 vectors in $$\real^{20}$$ span $$\real^{20}\text{?}$$

in-context