CS 530Spring 2000
E. Angel

Homework 1

1.

Strang 1.2.4 Sketch the three lines

x+2y=2

x-y=2

y=1

Can these three equations be solved simultaneously? What happens to the figure if all right hand sides are 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?

2.

Strang 1.2.11 The equations

ax+2y=0

2x+ay=0

are certain to have the solution x=y=0. For which values of ais there a whole line of solutions?

3.

Strang 1.2.12 Sketch the plane x+y+z=1, or the part of the plaane that is in the positive octant where $x\ge 0,$ $y\ge 0.$$z \ge 0.$ Do the same for x+y+z=2 in the same figure. What vector is perpendicular to those planes.

4.

Strang 1.29 (at end of chapter) Write down the 2 by 2 matrices which
(a) reverse the direction of every vector
(b) project every vector onto the x₂-axis
(c) turn every vector clockwise through 90 degrees
(d) reflect every vector through the 45 degree line x₁=x₂.

5.

Strang 2.1.2 Which of the following subsets of ${\bf R}^3$ are actually subspaces?
(a) The plane of vectors with first component b₁=0.
(b) The plane of vectors b with b₁=1.
(c) The vectors b with b₁b₂=0 (this is the union of the subspaces, the plane b₁=0 and the plane b₂=0.
(d) The solitary vector b=(0,0,0).
(e) All combinations of the two given vectors x=(1,1,0) and y=(2,0,1).
(f) The vectors (b₁,b₂,b₃) that satisfy b₃-b₂+3b₁=0.

6.

Strang 2.1.3 Describe the column space and the null space of the matrices

$$A=\left[\begin{array}{cc}1&-1\\ 0&0\end{array}\right] ~~{\rm ... ...eft[\begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&0\end{array}\right].$$

7.

Strang 2.1.5 (Note: inverses under multiplication are not necessary to define a set of scalars.) In the definition of a vector space, addition and scalar multiplication are required to satisfy the following rules:

(a)

x+y=y+x

(b)

x+(y+z)=(x+y)+z

(c)

There is a unique "zero vector" such that x+0=x for all x

(d)

For each x there is a unique vector -x such that x+(-x)=0

(e)

1x=x

(f)

(c₁c₂)x=c₁(c₂x)

(g)

c(x+y)=cx+cy

(h)

(c₁+c₂)x=c₁x+c₂x

(a) Suppose addition in ${\bf R}^2$ adds an extra one to each component, so that (3,1)+(5,0) equals (9,2) instead of (8,1). With scalar multiplication unchanged, which rules are broken?
(b) Show that the set of all positive real numbers with x+y and cxredefined to equal the usual xy and x^y, respectively, is a vector space. What is the "zero vector?"

8.

Can you form a set of scalars with 2, 3, or n elements? If so, show the operations and verify the rules are satisfied.

9.

A rational number is the ratio of two integers. Does the set of rational numbers under ordinary addition and multiplication form a set of scalars? Prove your result.

10.

Strang 2.3.1 Decide whether or not the following vectors are linearly independent by solving c₁v₁+c₂v₂+c₃v₃+c₄v₄=0

$$v_1=\left[\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right], v_... ...ht], v_4=\left[\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right].$$

Decide also if they span ${\bf R}^4$, by trying to solve c₁v₁+c₂v₂+c₃v₃+c₄v₄=(0,0,0,1)

11.

Strang 2.3.6 Describe geometrically, the suspace of ${\bf R}^3$spanned by
(a)(0,0,0), (0,1,0),(0,2,0)
(b) (0,0,1), (0,1,1),(0,2,1)
(c) all six of these vectors
(d) all vectors with positive components

12.

Strang 2.3.16 Find the dimension of the space of 3 by 3 symmetric matrices, as well as a basis.

13.

Stang 2.4.20 Write down a matrix with the required property or explain why no such matrix exists
(a) Column space contains $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right], \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],$ row space contains $\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 2\end{array}\right],$
(b) Column space has basis $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],$nullspace has basis $\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right].$
(c) Column space $={\bf R}^4$, row space $={\bf R}^3$.

14.

Can you form a vector space from n-tuples of binary numbers? Prove your result.

15.

Consider the recurrence

y_k = -3y_k-1 - 2y_k-2

Give two distinct bases for the vector space of solutions and give the matrices which convert a solution in one basis to a solution in the other.

Due: Monday, February 7.