9
$\begingroup$

In general, what is the most straightforward way to find the span of a set of vectors? I'm trying to find the span of these three vectors:

$$\{[1, 3, 3], [0, 0, 1], [1, 3, 1]\}$$

  • 0
    I would probably start by writing the set of vectors as a system of linear equations, then writing the system as an augmented matrix, and then converting to reduced row echelon form - is this the correct procedure?2012-12-07
  • 0
    Do you know the definition of span?2012-12-07
  • 0
    @Ockham Yes - the span of a set of vectors is the set of all linear combinations of a set of vectors. How can I find the set of all linear combinations of a set of vectors?2012-12-07
  • 1
    I think I understand it now (at least partially). The span would be a[1, 3, 3] + b[0, 0, 1] + c[1, 3, 1], which would be [a + c, 3a + 3c, 3a + b + c], where a, b, and c are arbitrary constants. But how can I use this information to determine whether the set of vectors spans R^3?2012-12-07
  • 0
    To simplify things a tad, write the matrix whose rows are the given vectors. Find a basis for the row space of this matrix by taking the non-zero rows of an echelon form. The vectors comprising this basis will have the same span as the set of original vectors. (Note you'll still be in the situation where you have to say "the span is the set of linear combinations of these vectors..."; but the aforementioned procedure will weed out any redundancies.)2012-12-07
  • 0
    That is correct2012-12-07
  • 0
    @DavidMitra What do you mean by "taking the non-zero rows?" Specifically, how can I find a basis by "taking" the rows?2012-12-07
  • 1
    Once you have the echelon form, the rows that are not identically zero (interpreted as vectors) will form the basis.2012-12-07

4 Answers 4