Upper triangular matrix as subspace

[trojansc82]

Homework Statement

Which of the following subsets of M_n,n are subspaces of M_n,n with the standard operations:

The set of all n x n upper triangular matrices

Homework Equations

10 axioms of vector space

The Attempt at a Solution

The set of all n x n upper triangular matrices is not closed under addition:

[ 1 1 1 ] [-1 -1 -1] [0 0 0]
[ 0 1 1 ] + [ 0 -1 -1] = [0 0 0]
[ 0 0 1] [0 0 -1] [0 0 0]

I apologize for the ugliness of the matrices, it is difficult to input a 3x3 upper triangular and demonstrate addition between two matrices.

 

[Mark44]

Homework Statement

Which of the following subsets of M_n,n are subspaces of M_n,n with the standard operations:

The set of all n x n upper triangular matrices

Homework Equations

10 axioms of vector space

The Attempt at a Solution

The set of all n x n upper triangular matrices is not closed under addition:

[ 1 1 1 [-1 -1 -1 [0 0 0
0 1 1 + 0 -1 -1 = 0 0 0
0 0 1] 0 0 -1] 0 0 0]

It took me a while to figure out what the above was supposed to mean. When I figured it out, I could see that what you thought was a counterexample actually isn't. An upper triangular matrix is a square matrix for which all the entries below the main diagonal are zero. The definition doesn't say anything about the entries above or on the main diagonal.

 

[trojansc82]

It took me a while to figure out what the above was supposed to mean. When I figured it out, I could see that what you thought was a counterexample actually isn't. An upper triangular matrix is a square matrix for which all the entries below the main diagonal are zero. The definition doesn't say anything about the entries above or on the main diagonal.

Oh ok great. So they can all be zero above or on the main diagonal as well.

 

[shelovesmath]

Homework Statement

Which of the following subsets of M_n,n are subspaces of M_n,n with the standard operations:

The set of all n x n upper triangular matrices

Homework Equations

10 axioms of vector space

The Attempt at a Solution

The set of all n x n upper triangular matrices is not closed under addition:


I apologize for the ugliness of the matrices, it is difficult to input a 3x3 upper triangular and demonstrate addition between two matrices.

Wow. That's a hideous matrix. Glad you figured it out though.

 

[Mark44]

Here's the LaTeX for one of your matrices. Click it to see the script.
\begin{bmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\end{bmatrix}