How many Hecke operators span the Hecke algebra?

"Span" question in Linear Algebra?

• I sort of understand what "span" mean.....My understanding is that if a augumented matrix has a solution. Ax=b's b part spans Matrix A But what I don't get is when the following question is asked. Determine if following Matrix spans R^4...what is THAT SPAN R^4 suppose to mean?!?!?! I know what R4 is but I need help on 1. what it it really means when u say "it spans R4" or not (prefer graphcially u can use R3 as example to explian that to me. ) 2. how to go about finding it. Here is the original Matrix [12 -7 11 -9 5 -9 4 -8 7 -3 -6 11 -7 3 -9 4 -6 10 -5 12 ] I have used excel to reduce it to [12 -7 11 -9 5 0 -55 11 11 33 0 0 -84 41 -122 0 0 0 0 -4 ] What's next?

• Answer:

  The span of a set of vectors is the set of all linear combinations of those vectors If you have 4 linearly independent vectors in R4, then they must span all of R4.