1 Vector Spaces
1.1 $R^{n}$ and $C^{n}$
1.1.1 Complex Numbers
As we have learned these things before, let's skip the content of this part.
1.1.2 Lists
- 1.1 notation: F
- Throughout this book, F stands for either R or C. The letter F is used because R and C are all example of what are called fields.
If we prove a theorem involving F, we will know that it holds when F is replaced with R or C.
Elements of F are called scalars. The word scalar is often used when we want to emphasize that an object is a number, as opposed to be a vector.
- 1.2 definition: list length
- Suppose $n$ is a nonnegative integer. A list of length $n$ is an ordered collection of 𝑛 elements (which might be numbers, other lists, or more abstract objects).
- Two lists are equal if and only if they have the same length and the same elements in the same order.
Lists are often written as elements separated by commas and surrounded by parentheses.A list of length 𝑛 might look like this: $(z_{1},\cdots,z_{n})$.
Remember, however, that by definition each list has a finite length that is a nonnegative integer. Thus an object that looks like (𝑥1,𝑥2,…), which might be said to have infinite length, is not a list.
A list of length 0 looks like this: ( ). We consider such an object to be a list so that some of our theorems will not have trivial exceptions.
Lists differ from sets in two ways: in lists, order matters and repetitions have meaning; in sets, order and repetitions are irrelevant.
1.1.3 $F^{n}$
- 1.3 definition: $F^{n}$, coordinate
- $F^{n}$ is the set of all lists of length 𝑛 of elements of $F: F^{n} = \{(x_{1},\cdots,x_{n}):x_{k}\in F ,k=1,\cdots,n\}$. For $(x_{1},\cdots,x_{n})\in F^{n}$ and $k=1,\cdots,n$ , we say that $x^{k}$ is the $k^{th}$ coordinate of $(x_{1},\cdots,x_{n})$.
- 1.4 definition: addition in $F^{n}$
- Addition in $F^{n}$ is defined by adding corresponding coordinates: $(x_1,\cdots,x_n)+(y_1,\cdots,y_n)=(x_1+y_1,\cdots,x_n+y_n)$ .
Often the mathematics of $F^{n}$ becomes cleaner if we use a single letter to denote a list of 𝑛 numbers, without explicitly writing the coordinates. For example, the next result is stated with $x$ and $y$ in $F^{n}$ even though the proof requires the more cumbersome notation of $(x_1,\cdots,x_n)$ and $(y_1,\cdots,y_n)$.
- 1.5 notation: 0
Let 0 denote the list of length 𝑛 whose coordinates are all 0
1.2 Definition of Vector Space
- 1.6 definition: addition, scalar multiplication
- An addition on a set $V$ is a function that assigns an element $u+v\in V$ to each pair of elements $u, v\in V$ .
- A scalar multiplication on a set $V$ is a function that assigns an element $\lambda v\in V$ to each $\lambda \in V$ and each $v\in V$.
Now we are ready to give the formal definition of a vector space.
- 1.7 definition: vector space
- A vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold.
- commutativity
- $u+v=v+u$ for all $u,v\in V$.
- associativity
- $(u+v)+w=u+(v+w)$ and $(ab)v=a(bv)$ for all $u,v,w \in V$ and for all $a,b\in F$.
- additive identity
- There exists an element $0\in V$ such that $v+0=v$ for all $v+w=0$.
- additive inverse
- For every $v\in V$, there exists $w\in V$ such that $v+w=0$.
- multiplicative identity
- $1v=v$ for all $v\in V$.
- distributive properties
- $a(u+v)=au+av$ and $(a+b)v=av+bv$ for all $a,b\in F$ and all $u,v\in V$.
- 1.8 definition: vector point
- Elements of a vector space are called vectors or points.
- 1.9 definition: real vector space, complex vector space
- A vector space over $R$ is called a real vector space.
- A vector space over $C$ is called a complex vector space.
- 1.10 notation: $F^S$
- If $S$ is a set, then $F^S$ denotes the set of functions from $S$ to $F$.
- For $f,g \in F^{S}$, the sum $f+g\in F^S$ is the function defined by $$(f+g)(x)=f(x)+g(x)$$for all $x\in S$.
The vector space $F^{n}$ is a special case of the vector space $F^{S}$ because each $(x_{1},\cdots,x_{n})\in F^{n}$ can be thought of as a function $x$ from the set {1,2,…,𝑛} to $F$ by writing $x(k)$ instead of $x_k$ for the $k$th coordinate of $(x_{1},\cdots,x_{n})$. In other words, we can think of $F^{n}$ as $F^{\{1,2,…,𝑛\}}$.
- 1.11 notation: -v, w-v
- Let $v,w\in V$. Then
- -v denotes the additive inverse of v;
w-v is defined to be w+(-v).
1.3 Subspaces
- 1.12 definition: subspace
- A subset $U$ of $V$ is called a subspace of $V$ if $U$ is also a vector space with the same additive identity, addition, and scalar multiplication as on V.
The next result gives the easiest way to check whether a subset of a vector space is a subspace.
- 1.13 conditions for a subspace
- A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions.
- additive identity
- $0\in U$.
- closed under addition
- $u,w \in U$ implies $u+w\in U$.
- closed under scalar multiplication
$a\in F$ and $u\in U$ implies $au\in U$.
1.3.1 Sums of Subspaces
- 1.14 definition: sum of subspaces
- Suppose $V_{1},\cdots,V_{n}$ are subspaces of $V$. The sum of $V_{1},\cdots,V_{n}$, denoted by $V_{1}+\cdots+V_{n}$, is the set of all possible sums of elements of $V_{1},\cdots,V_{n}$. More precisely, $$V_{1}+\cdots+V_{m}=\{v_{1}+\cdots+v_{m}:v_{1}\in V_{1},\cdots,v_{m}\in V_{m}\}$$
- 1.15 sum of subspaces is the smallest containing subspace
Suppose $V_{1},\cdots,V_{m}$ are subspaces of $V$. Then $V_{1}+\cdots+V_{m}$ is the smallest subspace of $V$ containing $V_{1},\cdots,V_{m}$.
Proof
It can verified that $V_{1}+\cdots+V_{m}$ contains the additive identity 0 and is closed under addition and scalar multiplication.So $V_{1}+\cdots+V_{m}$ is a subspace of $V$.
The subspaces $V_{1},\cdots,V_{m}$ are all contained in $V_{1}+\cdots+V_{m}$ (to see this, consider sums $v_{1}+\cdots+v_{n}$ where all except one of the $v_{k}$’s are 0). Conversely, every subspace of $V$ containing $V_{1},\cdots,V_{m}$ contains $V_{1}+\cdots+V_{m}$ (because subspaces must contain all finite sums of their elements). Thus $V_{1}+\cdots+V_{m}$ is the smallest subspace of 𝑉 containing $V_{1},\cdots,V_{m}$.1.3.2 Direct Sums
- 1.16 definition: direct sum, $\oplus$
- Suppose $V_{1},\cdots,V_{m}$ are subspaces of $V$.
- The sum $V_{1}+\cdots+V_{m}$ is called a direct sum if each element of $V_{1}+\cdots+V_{m}$ can be written in only one way as a sum $v_{1}+\cdots+v_{m}$, where each $v_{k}\in V_{k}$.
- If $V_{1}+\cdots+V_{m}$ is a direct sum, then $V_{1}\oplus\cdots\oplus V_{m}$ denotes $V_{1}+\cdots+V_{m}$, with the $\oplus$ notation serving as an indication that this is a direct sum.
example
$U=\{(x,y,0)\in F^{3}:x,y\in F\}$ and $W=\{(0,0,z)\in F^{3}:z\in F\}$, then $F^{3}=U \oplus W$. - 1.17 condition for a direct sum
- Suppose $V_{1},\cdots,V_{m}$ are subspaces of 𝑉. Then $V_{1}+\cdots+V_{m}$ is a direct sum if and only if the only way to write 0 as a sum $v_{1}+\cdots+v_{m}$, where each $v_{k}\in V_{k}$, is by taking each $v_{k}$ equal to 0.
Proof
First suppose $V_{1},\cdots,V_{m}$ is a direct sum. Then the definition of direct sum implies that the only way to write 0 as a sum $v_{1}+\cdots+v_{m}$, where each $v_{k}\in V_{k}$, is by taking each $v_{k}$ equal to 0.
Now suppose that the only way to write 0 as a sum $v_{1}+\cdots+v_{m}$, where each $v_{k}\in V_{k}$, is by taking each $v_{k}$ equal to 0. To show that $V_{1}+\cdots+V_{m}$ is a direct sum, let $v\in V_{1}+\cdots+V_{m}$. We can write$$v=v_{1}+\cdots+v_{m}$$
for some $v_{1}\in V_{1},\cdots,v_{m}\in V_{m}$. To show that this representation is unique, suppose we also have$$v=u_{1}+\cdots+u_{m},$$
where $u_{1}\in V_{1},\cdots,u_{m}\in V_{m}$. Subtracting these two equations, we have$$0=(v_{1}-u_{1})+⋯+(v_{m}-u_{m}).$$
Because $v_{1}-u_{1}\in V_{1},\cdots,v_{m}-u_{m}\in V_{m}$, the equation above implies that $v_{k}-u_{k}=0$. Thus $v_{1}=u_{1},\cdots,v_{m}=u_{m}$, as desired. - 1.18 condition for a direct sum of two subspaces
- Suppose $U$ and $W$ are subspaces of $V$. Then $U+W$ is a direct sum $\Leftrightarrow 𝑈\cap 𝑊 =\{0\}$.
Proof
First suppose that $U+W$ is a direct sum. If $v\in U\cap W$, then $0=v+(-v)$, where $v\in U$ and $-v\in W$ or $-v\in V$ and $v\in W$. By the unique representation of 0 as the sum of a vector in $U$ and a vector in $W$, we have $v=0$. Thus $𝑈\cap 𝑊 =\{0\}$, completing the proof in one direction.
To prove the other direction, now suppose $𝑈\cap 𝑊 =\{0\}$. To prove that $U+W$ is a direct sum, suppose $u\in U,w\in W$, and $$0=u+w.$$
We only need to show that $u=w=0$. The equation above implies that 𝑢 = −𝑤 ∈ 𝑊. Thus $u\in U\cap W$. Hence $u=0$, which by the equation above implies that $w=0$, completing the proof.
When asking about a possible direct sum with more than two subspaces, 1.18 is not enough to test that each pair of the subspaces intersect only at 0.
example
$V_{1}=\{(x,y,0)\in F^{3}:x,y\in F\}$
$V_{2}=\{(0,0,z)\in F^{3}:z\in F\}$
$V_{3}=\{(0,y,y)\in F^{3}:y\in F\}$