is the set of all upper triangular matrices a vector space?
I have tried to research if it's a vector space or a subspace but Linear Algebra is starting to look like foreign language for me. Can anyone help break this down and the difference between the two?
You need to verify that this subspace is closed under scalar multiplication and vector addition.
Since addition of matrices is done component-wise, any upper-triangular matrix sum will also be upper-triangular. (All that is needed is to note that the entries below the diagonal will consist of 0 + 0.)
Also, scalar multiplication applied to a matrix is distributed across each component. Since, a scalar times 0 produces zero, the entries below the diagonal will remain zero, and the result will remain upper-triangular.