1
$\begingroup$

I'm trying to determine whether [the set of all 2x2 matrices] is in the span of the following matrices:

1 0
0 1

0 1
0 0

0 0
1 0

0 0
0 1

If a basis for [the set of all 2x2 matrices] is in the span of these four matrices, then does the set of matrices span [the set of all 2x2 matrices]? Also, is there a faster way to determine whether the set spans [the set of all 2x2 matrices]?

  • 0
    I need to find a more concise notation for "the set of all 2x2 matrices" - the question above seems too verbose.2012-12-12
  • 0
    If you know the underlying ring, use matrix ring. For invertible matrices, use general linear group - $GL_n(\mathbb R)$2012-12-12
  • 0
    The standard definition of rings in abstract algebra - groups, rings, fields etc.2012-12-12
  • 1
    The set of all 2x2 matrices is usually denoted by $M_2(\mathbb{R})$ or $\mathbb{R}^{2\times 2}$.2012-12-12
  • 0
    @dexter04 Is there any way to find the underlying ring for the set of four matrices here?2012-12-12
  • 0
    There could be any kind of ring possible. It depends on whether you are dealing with reals, complex numbers, modular arithmetic or myriad other things. You need a ring to define an entry of the matrix. Going in reverse is useless. It is like defining a matrix without defining any entries.2012-12-12
  • 0
    Correction: The set of all 2x2 matrices over a field $\mathbb{F}$ is usually denoted by $M_2(\mathbb{F})$ or $\mathbb{F}^{2\times2}$.2012-12-12
  • 0
    @dexter04 I am referring to the set of 2x2 matrices of real numbers.2012-12-12
  • 0
    Then, you already have your ring, $(\mathbb R,+,.)$ - real numbers equipped with addition and multiplication2012-12-12
  • 0
    @dexter04 If a basis for [the set of all 2x2 matrices of real numbers] is in the span of these four matrices, then does the set of four matrices span [the set of all 2x2 matrices of real numbers]?2012-12-12

2 Answers 2

2

You are being asked whether it is true that every $2\times2$ matrix is a linear combination of the four matrices you are given. That is, you are being asked whether it is true that no matter what $a,b,c,d$ are you can find $r,s,t,u$ such that $$\pmatrix{a&b\cr c&d\cr}=r\pmatrix{1&0\cr0&1\cr}+s\pmatrix{0&1\cr0&0\cr}+t\pmatrix{0&0\cr1&0\cr}+u\pmatrix{0&0\cr0&1\cr}$$ When it's written that way, can you decide whether such $r,s,t,u$ exist? Can you, in fact, go even farther and find formulas for $r,s,t,u$ (in terms of $a,b,c,d$)?

  • 0
    This looks more like a question than an answer - does that make it a rhetorical answer?2012-12-12
  • 0
    It makes it an opportunity for you to have the joy of working out the details on your own.2012-12-12
  • 0
    I think the formulas would be a = r, b = s, c = t, and d = (r + u). In this case, it is clear that the matrices span the set of all 2x2 matrices. Does that mean that the four matrices are a basis for the set of all 2x2 matrices?2012-12-12
  • 0
    You have written the formulas for $a,b,c,d$ in terms of $r,s,t,u$. What you need is formulas for $r,s,t,u$ in terms of $a,b,c,d$, since you are trying to show that every matrix ["for every $a,b,c,d$"] is a linear combination of the four matrices ["there exist $r,s,t,u$"].2012-12-12
  • 0
    On the general case, hpw do you determine if a 2x2 matrix is in the span of a set of 2x2 matrices?2017-10-19
  • 0
    @Makogan, what you do is post that as a new question where everyone can see it, rather than hide it away as a comment on an old question.2017-10-19
  • 0
    I am way to used to the draconian ways of stack exchange when it comes to post questions, so I avoid it whenever I can2017-10-22
  • 0
    @Makogan, a million questions have been posted to m.se without incident, but, suit yourself.2017-10-22
1

The standard basis for all 2x2 matrices is: $$ \begin{matrix} 1 & 0 \\ 0 & 0 \\ \end{matrix} $$ $$ \begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix} $$$$ \begin{matrix} 0 & 0 \\ 1 & 0 \\ \end{matrix} $$$$ \begin{matrix} 0 & 0 \\ 0 & 1 \\ \end{matrix} $$

The first matrix in your problem $$ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} $$ is a linear combination of the the first and last matrices in the basis. So yes, the 4 given matrices are in the span of all 2x2 matrices.

  • 0
    Of course, the zero matrix is also a linear combination of the first and last matrices in the standard basis, so one needs to say a little more.2012-12-12
  • 0
    @ruwin Did you mean to write "all 2x2 matrices are in the span of the four given matrices." instead?2012-12-12
  • 0
    Yes, my mistake.2012-12-12