show that the set of all m-vectors whose first components is 0 (of R^m)
To show that the set of all m-vectors whose first component is 0 (of R^m) is a vector space, we need to show that it satisfies the vector space axioms:
1. Addition of vectors:
For any two vectors u and v in the set, their sum u + v can be defined as follows:
(u1, u2, ..., um) + (v1, v2, ..., vm) = (0, u2 + v2, ..., um + vm)
Since the first component of both u and v is 0, the first component of u + v is also 0.
2. Scalar multiplication:
For any scalar c and vector u in the set, the scalar multiple c*u can be defined as follows:
c * (u1, u2, ..., um) = (0, c * u2, ..., c * um)
Again, the first component of u is 0, so the first component of c*u is also 0.
3. Existence of a zero vector:
The zero vector is the vector where all components are 0. In this set, the zero vector is (0, 0, ..., 0).
4. Existence of additive inverses:
For any vector u in the set, its additive inverse -u can be defined as follows:
-(u1, u2, ..., um) = (0, -u2, ..., -um)
Again, the first component of u is 0, so the first component of -u is also 0.
Therefore, the set of all m-vectors whose first component is 0 is closed under addition, scalar multiplication, contains a zero vector, and every vector has an additive inverse. Thus, it is a vector space.