Prove that u + v = v + u for any u and v in R^n.
First note that for `a, b \in R` , `a+b = b+ a` . This is the commutative property of addition (scalar).
Suppose `u, v \in R^n` are n-dimensional vectors such that:
`u = <u_1, u_2, ..., u_n>` and
`v = <v_1, v_2, ..., v_n>` .
By definition of vector addition:
`u + v = <u_1 + v_1, u_2 + v_2, ..., u_n + v_n>`
By the commutative property of addition,
`u + v = <u_1 + v_1, u_2 + v_2, ..., u_n + v_n> `
`= <v_1 + u_1, v_2 + u_2, ..., v_n + u_n> = v + u.`
Therefore, for any n-dimensional vectors u and v, u + v = v + n.