Final answer:
The dimension of a vector space V in three-dimensional space is 3, consisting of three basis unit vectors î, â, and ê. A vector in V is expressed using these unit vectors in component form. Vector operations like addition and scalar multiplication are commutative.
Step-by-step explanation:
To find the dimension of the vector space V, first we need to know how many basis vectors can span the entire space without redundancy. In a three-dimensional space, the dimension of the space is 3, which corresponds to the number of unit vectors necessary to describe any vector in this space. These unit vectors are often denoted as î (for the x-axis), â (for the y-axis), and ê (for the z-axis). As the standard Cartesian coordinate system is used to define a basis, these unit vectors would constitute a basis for V.
A vector in three-dimensional space is expressed in component form relative to these unit vectors as V = xi + yj + zk, where x, y, and z are the scalar components along the respective axes. The properties of vector addition and scalar multiplication in vector spaces assure us that any vector in V can be represented as a linear combination of the three unit vectors, confirming a three-dimensional space. Therefore, a basis for V would be {î, â, ê}.
Vector operations such as addition and scalar multiplication follow simple algebraic rules and can be done in any order because these operations are commutative in nature. Moreover, understanding these vector operations is essential for applications in fields such as mechanics and electromagnetism, which are explored through Euclidean vectors in physics.