Linear Algebra and Proving a Subspace

Date: 02/04/2004 at 06:07:27
From: Pete
Subject: Linear Algebra

(a) The set Sm = {(2a,b - a,b + a,b) : a,b are real numbers.  Under
the operations of matrix addition and multiplication, prove that this
is a subspace of M22.

(b) Find a basis for this subspace and give the dimension of the 
subspace.

Date: 02/05/2004 at 11:24:18
From: Doctor Jordan
Subject: Re: Linear Algebra

Hi Pete,

(a) Recall that M22 is the set of all 2x2 matrices with real entries, 
thus it is spanned by matrices,

  M22 = [q r] = q[1 0] + r[0 1] + s[0 0] + t[0 0].
        [s t]    [0 0]    [0 0]    [1 0]    [0 1]

Note that M22 is a vector space (it can be shown to satisfy the 10 
axioms of being a vector space).  As well, a basis B of M22 is the 
above set of four matrices.  Thus [M22]B, the coordinates of M22 with 
respect to basis B, is [q].
                       [r]
                       [s]
                       [t]

Sm can be represented with vectors,

  [2a ] = a[2 ] + b[0]
  [b-a]    [-1]    [1]
  [b+a]    [1 ]    [1]
  [b  ]    [0 ]    [1]

As any vector representing Sm can be expressed like [M22]B for some q, 
r, s, t in the real numbers, Sm is a subspace of M22.

(b) Note that when we represent Sm as a vector, it becomes a linear 
combination of two vectors with weights a and b in the reals.  Thus Sm 
can be represented as the span of these two vectors,

  Span{[2 ], [0]}.
       [-1], [1]    
       [1 ]  [1]
       [0 ]  [1]

It can be seen that the vectors in this spanning set are linearly 
independent.  As they span Sm and are linearly independent, they are a 
basis for Sm.  We see that dim{Sm} = 2, as there are two vectors in
Sm's basis.

I hope this has helped you.  Looking at matrices like vectors is often 
useful, as it allows us to more easily consider linear dependence and 
spanning.  If anything I've said here does not make sense, or if you 
have any other questions, please write me back.

- Doctor Jordan, The Math Forum
  http://mathforum.org/dr.math/