# Subspace of a 4x4 matrix space

Science Advisor
Homework Helper

## Homework Statement

This question is taken from Linear Algebra Done Wrong by Treil. Question 7.5 of chapter 1 says this:
What is the smallest subspace of the space of 4 4 matrices which contains all upper triangular matrices (aj,k = 0 for all j > k), and all symmetric matrices (A = AT )? What is the largest subspace contained in both of those subspaces?

## Homework Equations

May be these aj,k = 0 for all j > k and A = AT

## The Attempt at a Solution

I'm just not sure with my answer because it sounds too short for such an elaborate question.
1st answer: the zero matrix, the one with all-zero entries.
2nd answer: the two subspaces themself.

## Answers and Replies

RUber
Homework Helper
There maybe some grammatical ambiguity.
I read this to say, let U be the space of 4x4 matrices, V be the space of upper triangular 4x4 matrices and W be the space of symmetric 4x4 matrices.
What is the smallest subspace of U which contains V and W? To this question, the smallest subspace still has to contain all the upper triangular and symmetric matrices. So it cannot be smaller than the Union of the two.
The largest subspace in the intersection of U, V, W would be a subspace which has all possible 4x4, symmetric, upper triangular matrices.

Otherwise--the smallest subspace of any space is the 0 space.

The largest space question seems clear either way. So I ask, what type of upper triangular matrices are still upper triangular in their transposes?

Science Advisor
Homework Helper
But the zero matrix is an element in both V and W, using your notation. And I thought of that answer of mine referring to a term defined in the mentioned book as "trivial subspace" which consists of the vector space itself and {0} (the zero vector only). So I based on the fact that zero vector has its own name that's why I choose that answer.

The largest subspace in the intersection of U, V, W would be a subspace which has all possible 4x4, symmetric, upper triangular matrices.
So by this you mean all matrices which are symmetric and upper triangular at the same time, which means the diagonal matrices? In that case it seems like the largest subspace contains "smaller" number of elements, well I know I can't really say "smaller" since there are infinite number of possibilities. I mean it's stricter than the smallest subspace.
Is there something I understand it wrong?

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
IDK about Linear Algebra done wrong, but this thread is off-topic for an Intro Physics HW forum.

I'm moving it the the Calculus and Beyond HW forum.

RUber
Homework Helper
I interpret the first question to be asking for a subspace X of U such that V and W are subspaces of X. The trivial subspace does not contain V and W.
For the second, I think diagonal is right.

Science Advisor
Homework Helper
Ah sorry, I'm fine with that.