Complete the following tasks using your personal copy of Sage or one of the servers (notice: the PLU server seems to be working well and has thus far been a stable option).
1. Solve the equation $x^2-x-1=0$ and then find the roots of the cubic equation $x^3+x^2+x+1$
2. Solve the following system of equations: $2x+y+z=1$, $3x+2y-z=0$, and $4x+3y+5z=4$.
- Plot the the previous three equations in $\mathbb{R}^3$ using the implicit_plot3d command.
3. Plot the system of equations given by $3x+2y+2z=1$ and $6x+4y+4z=0$ (each given implicitly). Based on this plot, do you think there will be a solution to the system? Why or why not?
4. Solve the following system of equations: $3x+y=1$, $6x+2y=2$ and $x-2y=3$. Explain briefly the answer you get; you may also want to use the graphical representation of this system to describe the answer.
5. Define vectors v=(2,4,5,6,2,1) and u=(4,7,2,8,9,-3). Find the linear combination 7v+8u.
- Provide 3 more vectors that are linear combinations of u and v. Find one vector that is not a linear combination of u and v.
- Determine if (5,6,1,2,1,4) and (6,10,-1,10,16,-7) are in Span(u,v). If either one is, prove it by finding the appropriate coefficients (either by hand or using Sage).
6. Define a procedure called projection(u,v) that returns the projection of $\mathbf{u}$ on to $\mathbf{v}$ (i.e. that computes $\mbox{proj}_\mathbf{v}\mathbf{u}$)
- Use your projection function to find 3 projections of your choosing, each in a different dimension.
- What happens if you evaluate projection(u,0)? Is this a problem?
7. Define a procedure called angle(u,v) that will calculate the angle between two vectors (any vectors).
- For the projections you computed above, find the angle between the vectors (so there should be 3).
8. Explore: Create the span of (1,2,3,4), (5,6,7,8), (9,10,11,12), and (13,14,15,16). Do you need all four of these vectors for the span? Is there a smaller collection of vectors that gives rise to the same span? Hint: create the span in Sage and see what Sage says about this span. Try to interpret the result.