Exercise 7

Provide a justification for your response to the following questions.

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

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

3. Do the vectors $$\evec_1=\threevec{1}{0}{0}, \evec_2=\threevec{0}{1}{0}, \evec_3=\threevec{0}{0}{1}$$ span $\real^3$?

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

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