Peter Selinger: Math 1341B, Questions and Answers

Math 1341B, Introduction to Linear Algebra
Questions and Answers

Questions and Answers

I often receive math questions from students. Since the answers might be useful to everybody, I am posting them here. I do this anonymously so as not to embarrass the student who asked the question. Ask a question by sending me email at selinger@mathstat.dal.ca (please mention 1341 in the subject line).

Spanning Sets

Question (Sep. 29): I was wondering if there was an easy way to prove that vectors form a spanning set. For example, one question from one of the previous exams says something along the lines of: Is it true that {(1,0) , (1,1)} spans the two-dimensional real numbers? Is there some way to prove this or is it simply a matter of being able to see that this is true?

Answer: you must show that for any vector (x,y), there exist scalars a,b such that

 (x,y) = a(1,0) + b(1,1).

This can be done by solving it for a and b. For instance, in this example, we get b=y, a=x-y.

If the vectors do not form a spanning set, e.g. {(1,2), (2,4)}, then the above does not have a solution, e.g.

 (x,y) = a(1,2) + b(2,4)

cannot be solved for a and b (try!). For now, this is the method you will have to use. Later, we will learn better methods.

Triangular form, Echelon form, row canonical form

Question (Nov. 5): As I was studying for the test I became quite confused. My question is, are: row canonical form, Echelon form, triangular form, and row reduced form all refering to the same thing? E.i. to the following matrix?

Answer: No, they don't refer to the same thing.

Triangular form: p.67
Triangular form only applies to square (i.e. n x n) matrices. It means that there are no entries below the diagonal.
Echelon form: p.68 and p.73
Echelon form applies to arbitrary (n x m) matrices. It means that the first non-zero entry in each row is to the right of the first non-zero entry in the previous row.
Row canonical form (also sometimes called row reduced form): p.74
A row canonical form is an Echelon form with these additional properties: the pivot entries are 1, and each pivot entry is the only non-zero entry in its column.

Examples:

  1 2 3           triangular: no
  1 0 1           Echelon: no
  0 0 1           row canonical: no
  
  1 2 3           triangular: yes
  0 0 1           Echelon: no
  0 0 1           row canonical: no
  
  1 2 3           triangular: yes
  0 2 1           Echelon: yes
  0 0 1           row canonical: no
  
  1 0 0           triangular: yes
  0 2 0           Echelon: yes
  0 0 1           row canonical: no
  
  1 0 0           triangular: yes
  0 1 0           Echelon: yes
  0 0 1           row canonical: yes
  
  1 0 2           triangular: yes
  0 1 2           Echelon: yes
  0 0 0           row canonical: yes
  
  1 2 3 4         triangular: n/a (no)
  0 0 1 5         Echelon: no
  0 0 1 6         row canonical: no
  
  1 2 3 4         triangular: n/a (no)
  0 0 1 5         Echelon: yes
  0 0 0 0         row canonical: no
  
  1 2 0 4         triangular: n/a (no)
  0 0 2 5         Echelon: yes
  0 0 0 0         row canonical: no
  
  1 2 0 4         triangular: n/a (no)
  0 0 1 5         Echelon: yes
  0 0 0 0         row canonical: yes

I hope these are enough examples to illustrate all 3 concepts.

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Peter Selinger / Department of Mathematics and Statistics / Dalhousie University
selinger@mathstat.dal.ca / PGP key