Unit Lower Triangular Matrices as Vector Spaces | Proof & Properties

[Maxwhale]

Homework Statement

Is the following a vector space?

The set of all unit lower triangular 3 x 3 matrices

[1 0 0]
[a 1 0]
[b c 1]

Homework Equations

Properties of vector spaces

The Attempt at a Solution

I checked the properties of vector space (usual addition and scalar multiplication). I proved that the upper triangular 3 x 3 matrices are vector spaces as they suffice all the properties of vector space. But I am not quite confident about lower triangle.

 

[Mark44]

Did you verify that upper triangular 3 x 3 matrices satisfy all 10 of the axioms for a vector space? From what you wrote you might be thinking that there are only two axioms.

OTOH, if all you need to do is prove that the set of upper triangular (or lower triangular) 3 x 3 matrices is a subspace of the vector space of 3 x 3 matrices, then all you need to do is show that both of the following are true:

1. If A and B are lower triangular 3 x 3 matrices, then so is A + B.
2. If A is any lower triangular 3 x 3 matrix, and c is a scalar, then c*A is a lower triangular 3 x 3 matrix.

As a shortcut, if you can show that c*(A + B) is lower triangular 3 x 3, with A, B, and c as described above, that will do it.

If you're actually trying to show that these lower triangular matrices form a vector space, which of the 10 axioms are you having trouble with?

 

[HallsofIvy]

"Lower triangular matrices" have exactly the same properties as "upper triangular matrices" and the proofs are essentially the same. But the crucial word here is "unit". I assume the problem assumes the usual addition and scalar multiplication of matrices. Is the 0 matrix in the set of "unit lower triangular matrices"?