## Exercises

For the eeauations $x + y = 4, 2x - 2y = 4$, draw the row picture (two intersecting lines) and the column picture (a combination of two columns, equatl to the column vector $(4, 4)$ on the right side).

Solve the non singular triangular system: $\begin{aligned} u + v + w &= b_1 \\ v + w &= b_2 \\ w &= b_3 \end{aligned}$ Show that your solution gives a combination of columns that equal the column on the write

Describe the intersection of three planes $u + v + w + z = 6$ and $u + w + z = 4$ and $u + w = 2$ (all in 4-dimensional space). Is it a line or a point or an empty set? What is the intersection of the fourth plane u = -1 is excluded?

Sketch the lines $\begin{aligned} x + 2y &= 2\\ x - y &= 2\\ y &= 1 \\ \end{aligned}$ Can the three equations be solved simultaneously? What happens to the ficutre if all right hand sides zero? Is there any nonzero choice of right hand sides which allows the three lines to intersect at the same point and the three equations to have a solution?

Find two points on the line intersection of three plances $t=0$, and $z=0$ and $x+y+z+y=1$ in four-dimensional space.

When $b=(2,5,7)$, find a solution $(u, v w)$ to equation (4) other than the solutions $(1, 0 1)$ and $(4, -1, -1)$, metnioned in the text.

- Equation 4: $u \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} v \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} w \begin{bmatrix} 1 \\ 3 \\ 4 \end{bmatrix} = b$

Give two more righthand sides in addition to $b=(2,5,7)$ for which equation (4) can be solved. Give two more right hand sides in addition to $b=(2,5,6)$ for which it cannot be solved.

Explain why the system: $\begin{aligned} u + w + w &= 2\\ u + 2v + 3w &= 1\\ v + 2w &= 0 \\ \end{aligned}$ if singular, by finding a combination of three equations that adds up $0 = 1$. What value should replace the last zero on the rightside to allow the equation to have solutions — and what is one of the solutions?

The column picture for the prevous exercises is $u \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} v \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} w \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} = b$ Show that the three columns on the left lie in the same plane by expressing the third column as a combination of the first two. What are the solutions $(u, v, w)$, if b is the zero vector $(0, 0, 0)$?

Under what condiction on $y_1, y_2, y_3$, do the points $(0, y_1), $(1, y_2)$, (2, y_3)$ lie on a straight line?

The equations $\begin{aligned} ax + 2y &= 0\\ 2x + ay &= 0 \\ \end{aligned}$ are certain to have the solution $x=y=0$. For which values of $a$, is there a whole line of solutions?

Sketch the plane $x+y+z=1$, or the part of the plane that is in the positivie octant where $x\geq0, y\geq0, z\geq0$. Do the same for $x+y+z=2$ in the same figure. What vector is Perpendicular to does planes?

Starting with the line $x+4y=7$, fin d the equation for the parallel line thruough $x=0, y=0$. Find the equation of another line that meets the first at $x=3, y=1$