Exercise 3

Consider the following vectors in $\real^4$: $$\vvec_1 = \fourvec{1}{1}{1}{1}, \vvec_2 = \fourvec{0}{1}{1}{1}, \vvec_3 = \fourvec{0}{0}{1}{1}, \vvec_4 = \fourvec{0}{0}{0}{1}$$.

1. Explain why $\bcal=\{\vvec_1,\vvec_2,\vvec_3,\vvec_4\}$ forms a basis for $\real^4$.

2. Explain how to convert $\coords{\xvec}{\bcal}$, the representation of a vector $\xvec$ in the coordinates defined by $\bcal$, into $\xvec$, its representation in the standard coordinate system.

3. Explain how to convert the vector $\xvec$ into $\coords{\xvec}{\bcal}$, its representation in the coordinate system defined by $\bcal$.

4. If $\xvec=\fourvec{23}{12}{10}{19}$, find $\coords{\xvec}{\bcal}$.

5. If $\coords{\xvec}{\bcal}=\fourvec{3}{1}{-3}{-4}$, find $\xvec$.