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## Basics of Vector Spaces

Definition. A real vector space is a nonempty set $$V$$ of objects, called vectors, with two operations, viz, addition and scalar multiplication, that satisfy the following properties for all vectors $$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$$ in $$V$$ and all scalars (real numbers) $$c$$ and $$d$$.

1. $$\overrightarrow{u}+\overrightarrow{v}$$ is in $$V$$.

2. $$\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}$$

3. $$(\overrightarrow{u}+\overrightarrow{v})+\overrightarrow{w}=\overrightarrow{u}+(\overrightarrow{v}+\overrightarrow{w})$$

4. There is a zero vector $$\overrightarrow{0}$$ such that $$\overrightarrow{u}+\overrightarrow{0}=\overrightarrow{u}$$.

5. There is a vector $$-\overrightarrow{u}$$ such that $$\overrightarrow{u}+(-\overrightarrow{u})=\overrightarrow{0}$$.

6. $$c\overrightarrow{u}$$ is in $$V$$.

7. $$c(\overrightarrow{u}+\overrightarrow{v})=c\overrightarrow{u}+c\overrightarrow{v}$$

8. $$(c+d)\overrightarrow{u}=c\overrightarrow{u}+d\overrightarrow{u}$$.

9. $$c(d\overrightarrow{u})=(cd)\overrightarrow{u}$$.

10. $$1\overrightarrow{u}=\overrightarrow{u}$$.

Remark.

1. Scalars are elements of a field such as the set of real numbers and the set of complex numbers. If scalars are complex numbers, then $$V$$ is called a complex vector space.

2. From the definition we have the following:
1. $$0\overrightarrow{u}=\overrightarrow{0}$$

2. $$c\overrightarrow{0}=\overrightarrow{0}$$

3. $$-\overrightarrow{u}=(-1)\overrightarrow{u}$$

Example. The following are real vector spaces.

1. $$V_n$$, the set of all vectors (directed line segments) in $$\mathbb R^n$$.

• Scalar multiplication: Usual scalar multiplication of vectors.

2. $$\mathbb R^n$$ and $$\mathbb C^n$$.

• Scalar multiplication: Entrywise scalar multiplication.

3. $$\mathbb R^{\infty}$$, the set of all real sequences $$(a_n)=(a_1,a_2,a_3,\ldots)$$.

• Scalar multiplication: Entrywise scalar multiplication.

4. $$P_n$$, the set of all real polynomials of degree at most $$n$$.

• Addition: If $$\overrightarrow{p}(t)=a_0+a_1t+\cdots+a_nt^n$$ and $$\overrightarrow{q}(t)=b_0+b_1t+\cdots+b_nt^n$$, then $(\overrightarrow{p}+\overrightarrow{q})(t)=(a_0+b_0)+(a_1+b_1)t+\cdots+(a_n+b_n)t^n.$
• Scalar multiplication: If $$\overrightarrow{p}(t)=a_0+a_1t+\cdots+a_nt^n$$ and $$c\in \mathbb R$$, then $(c\overrightarrow{p})(t)=ca_0+ca_1t+\cdots+ca_nt^n.$
5. $$F$$, the set of all real-valued functions on a set $$D$$.

• Addition: $$(\overrightarrow{p}+\overrightarrow{q})(x)=\overrightarrow{p}(x)+\overrightarrow{q}(x)$$ for all $$\overrightarrow{p}, \overrightarrow{q}\in F$$.

• Scalar multiplication: $$(c\overrightarrow{p})(x)=c\overrightarrow{p}(x)$$ for all $$\overrightarrow{p}\in F$$ and $$c\in \mathbb R$$.

6. $$L(V,W)$$, the set of all linear transformations $$T:V\to W$$ where $$V$$ and $$W$$ are real vector spaces.

• Addition: $$(\overrightarrow{T}+\overrightarrow{S})(\overrightarrow{v})=\overrightarrow{T}(\overrightarrow{v}) +\overrightarrow{S}(\overrightarrow{v})$$ for all $$\overrightarrow{v}\in V$$.

• Scalar multiplication: $$(c\overrightarrow{T})(\overrightarrow{v})=c\overrightarrow{T}(\overrightarrow{v})$$ for all $$\overrightarrow{v}\in V$$ and $$c\in \mathbb R$$.

7. $$M_{m,n}(\mathbb R)$$, the set of all $$m\times n$$ real matrices.