How to prove that the addition and scalar multiplication of all upper triangular matrices are linear spaces in real number field

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