How to prove that the addition and scalar multiplication of all upper triangular matrices are linear spaces in real number field
V = {a | a upper triangular matrix}
Because of the properties of matrix addition and scalar multiplication, the properties of linear operation are not self-evident
As long as it is proved that:
Closure of addition and scalar multiplication
1) A, B ∈ V, upper triangular matrix + upper triangular matrix is still upper triangular matrix, so a + B ∈ v
2) A ∈ V, scalar multiplication λ A is upper triangular matrix, λ a ∈ v
Existence of zero elements: 0 matrix is upper triangular matrix
1)A+0=A
2) A+(-A)=0