Vector Spaces

In this chapter, we present basic notions of group, ring, field, and then vector space. The concept of field leads us to the familiar entities: ℚ $${\mathbb {Q}}$$ , ℝ $$\mathbb {R}$$ , and ℂ $${\mathbb {C}}$$ . A vector space over a field is a set together with vector addition, defined on the product of the vector space with itself, and scalar multiplication, defined on the product of the vector space and the field, that further satisfies a list of conditions for scalar multiplication and vector addition. On a vector space, we can do linear algebra via vector addition and scalar multiplication. A vector space reminds us of ℝ n $$\mathbb {R}^n$$ or ℝ n 1 × n 2 × ⋯ × n p $$\mathbb {R}^{n_1 \times n_2 \times \cdots \times n_p}$$ (but with less technical capabilities). We can define linear operators of vector space to vector space. The concept of linear combination, subspaces, product space, linear variety, linear independence, and dimension is introduced in vector spaces. For real or complex vector spaces (vector spaces over the field ℝ $$\mathbb {R}$$ or ℂ $${\mathbb {C}}$$ ), the concept of convexity can be defined.