[ 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]
trojansc82 said:Homework Statement
Which of the following subsets of Mn,n are subspaces of Mn,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:
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.[ 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]
Mark44 said: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 said:Homework Statement
Which of the following subsets of Mn,n are subspaces of Mn,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.
An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero. This means that all the non-zero elements are located on or above the diagonal from the top left to the bottom right.
An upper triangular matrix is typically represented as a square array with rows and columns. The elements below the main diagonal are usually left blank or filled with zeros. The non-zero elements are written in the upper right corner of the matrix.
An upper triangular matrix is important in linear algebra because it can be used to solve systems of equations, calculate determinants, and perform other operations. It also has a simpler structure compared to a general matrix, making it easier to work with and compute.
An upper triangular matrix can be considered as a subspace of a larger vector space. It satisfies the three properties of a subspace: closure under addition, closure under scalar multiplication, and contains the zero vector. This allows for easier analysis and manipulation of the matrix.
An upper triangular matrix is a subspace only if all the elements on the main diagonal are non-zero. If any element on the main diagonal is zero, it does not satisfy the three properties of a subspace and therefore cannot be considered as a subspace. Additionally, the dimension of the subspace is limited by the size of the matrix, as it can only contain as many linearly independent vectors as there are rows/columns in the matrix.