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I am trying to solve the following exercise,

What is the smallest subspace of 3 by 3 matrices that contains all symmetric matrices and all lower triangular matrices? Moreover, what is the largest subspace that is contained in both of those subspaces?

I would be very grateful if you could help me to understand in any way, either with a hint or an analytical explanation, to the following:

  • Is there a standard definition of a subspace so as to describe mathematically the initial space of $3 \times 3$ matrices and the contained subspaces that formed by the lower triangular and symmetric matrices?
  • Given that the space of $3 \times 3$ matrices contains all symmetric matrices and all lower triangular matrices,
    • Can we write this literal explanation in mathematical terms using matrices?
    • What is the intuitive explanation?
  • Can we prove that the space of $3 \times 3$ matrices is closed under addition and scalar multiplication?
  • What methodology should we follow so as to compute the smallest and largest $3\times3$ subspace?
  • Finally, is there a geometrical explanation to this problem?

Thank you!!

If that matters, I would like to note that this is not an exercise for a course but for a personal pursue:)

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    $\begingroup$ Let $ V $ be a vector space. A subspace of $ V $ is a subset of $ V $ which is closed under addition and scalar multiplication. Note that a subspace of $ V $ is itself a vector space. $\endgroup$ – littleO Dec 4 '15 at 20:25
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    $\begingroup$ ... but not every subset of $V$ which is closed under addition and scalar multiplication is a subspace (littleO is leaving out one important axiom). $\endgroup$ – Michael Albanese Dec 4 '15 at 20:33
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    $\begingroup$ @Michael Albanese the axiom is that the zero vector in our case the zero matrix should belong to the vector space? $\endgroup$ – darkmoor Dec 4 '15 at 20:36
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    $\begingroup$ @MichaelAlbanese Good point, I forgot to say nonempty subset. $\endgroup$ – littleO Dec 4 '15 at 20:44
  • $\begingroup$ @darkmoor: Exactly. $\endgroup$ – Michael Albanese Dec 4 '15 at 20:45
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Recall that a vector subspace is a nonempty subset $U$ of a vector space $V$ that is itself closed under the addition and scalar multiplication operations that it inherits from $V$. So really, it can be considered as a vector space itself in its own right.

Recall that the set of all linear combinations of elements in a set $S$ is called the span of that set, $\operatorname{Span}(S)$. The span of a set is the smallest subspace that contains that set. Thus we can rephrase the problem as asking: What is the span of the set of all lower triangular and symmetric $3\times3$ matrices?

This problem is simplified by the fact that the set of all lower triangular $(3\times3)$ matrices and the set of all symmetric $(3\times3)$ matrices are each subspaces by themselves (you should prove this). Therefore, given any linear combination of lower triangular and symmetric matrices, we can collect all the lower triangular matrices into a single lower triangular matrix, and all the symmetric matrices into a single symmetric matrix. So the subspace spanned by these sets can be seen to be the set of all matrix sums $A+B$, where $A$ is lower triangular and $B$ is symmetric. Now here's a big hint: Given any $3\times3$ matrix $M$, can you write it in such a form? Hopefully you can take it from there.

EDIT: I would like to elaborate here on the second part of the question, which was concerning the largest subspace contained in the two subspaces. I missed that part the first time around, and when you asked me about it in the comments, I misunderstood and gave you an incorrect response. I am terribly sorry for causing you further confusion.

Regarding the largest subspace contained in the two subspaces, it seems that the most obvious candidate should be the intersection of the two spaces. After all, it is the largest set contained in the two subspaces. And a subspace it is indeed. (If you haven't done this already, you should prove that the intersection of two subspaces is a subspace). So you should ask, if a matrix is contained in the intersection of the two subspaces, i.e., it is both lower triangular and symmetric, what would it look like? That should lead you to the answer.

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  • $\begingroup$ Thanks for answering, I was thinking your third paragraph, why we can collect all the lower triangular matrices into a single lower triangular matrix, and all the symmetric matrices into a single symmetric matrix? Can you explain it a litle more? $\endgroup$ – darkmoor Dec 4 '15 at 21:21
  • $\begingroup$ @darkmoor The observation that the set of lower triangular matrices is a subspace means that a linear combination of lower triangular matrices is a lower triangular matrix. The same goes for symmetric matrices $\endgroup$ – silvascientist Dec 4 '15 at 21:22
  • $\begingroup$ Helpful. Also, why do you write the span just as $\mathbf{A} + \mathbf{B}$ and not with linear coefficients like this $\alpha\mathbf{A} + \beta \mathbf{B}$? $\endgroup$ – darkmoor Dec 4 '15 at 21:28
  • $\begingroup$ @darkmoor Because $\alpha A$ is just another lower triangular matrix, and $\beta B$ is just another symmetric matrix. $\endgroup$ – silvascientist Dec 4 '15 at 21:37
  • $\begingroup$ So what you are suggesting is to prove that $\mathbf{M} \equiv \mathbf{A} + \mathbf{B}$? $\endgroup$ – darkmoor Dec 4 '15 at 21:44

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