Chapter 3 Vector Space Objective: To introduce the notion of vector space, subspace, linear independence, basis, coordinate, and change of coordinate.

Recall, In - vector addition - scalar multiplication - norm - triangle inequality §3-1 Definition and Examples

Why Introduces Vector Space?  It provides comprehensive understanding of many mathematical & physical phenomena. For example, All the solutions of the ODE can be described as. Why? Controllability and observability space in linear control theory. m

Vector Space Axioms Definition: Let be set and be a field ( in most practical case, ). Define two binary operations Then is a vector space if the follow- ing Conditions hold:

Vector Space Axioms (cont.) For any, A1: A2: A3: A4:

Vector Space Axioms (cont.) A5: A6: A7:

Examples  defined by over is a vector space.

Examples (cont.)  over is also a vector space with defined by (1) and (2).  over is a vector space.  over is NOT a vector space. (Why?)

Examples (cont.)  Let  over defined by is a vector space. 

Examples (cont.)  defined by is a vector space.  is NOT a vector space. (Why?)

Theroem3.1.1: Let be a vector space and. Then PF:

Definition: If is a nonempty subset of a vector space, and satisfies the following conditions: then is said to be a subspace of. §3-2 Subspace Remark 1: Thus every subspace is a vector space in its own right. Remark 2: In a vector space, it can be readily verified that and are subspaces of. All other subspaces are referred to as proper subspaces.

Examples of Subspaces Example 2. (P.135)

Examples of Subspaces (cont.) Example 3. (P.135) Example 4. (P.135)

Examples of Subspaces (cont.) Example 5. (P.136) Example 8. (P.136) Example 6. (P.136)

Nullspace and Range-space  Let, ※ Define that N(A) is called the nullspace of A; R(A) is called the range ( column ) space of A.

Examples of Nullspaces Example 9. (P.137) Question: Determine N(A) if. Answer:

Note Note that, both the vector spaces and the solution set of contain infinite number of elements. Question: Can a vector space be described by a set of vectors with number being as small as possible? Example: Spanning set, linear independent, basis

Span and Spanning Sets Definition: Let be vectors in a vector space, a sum of the form, where are scalars, is called linear combination of. Definition: Definition: is said to be a spanning set for if

Examples of Span Example :

Theroem3.2.1: If, then is a subspace of. Question: Given a vector space and a set, how to determine whether or not?

Example 11. (P.140) Yes, Yes, let ∵ A is nonsingular, The system has a unique solution

Example 11.(c) (P.141) No,

Example 12. (P.141) Yes, let

Question: How to find a minimal spanning set of a vector space (i.e. a spanning set that contains the smallest possible number of vectors.) (i.e. There is no redundancy in a spanning set.) §3-3 Linear Independence It’s unnecessary.

Linear Dependency Definition: is said to be linear independent if “ ”. Definition: is said to be linear dependent if there exist scalars NOT all zero such that

Lemma :

Note 1: Linear independency means there is no redundancy on the spanning set. Note 2: is a minimal spanning set for iff is linear independent and spans. Definition: A minimal spanning set is called a basis.

Linear Dependency (cont.) Question: How to systematically determine the linear dependency of vectors ? Geometrical interpretation (see Figure 3.3) :

Example 3. (P.149) Note that is redundant for the spanning set. On the other hand, ∵ A is singular det(A)=0. a nontrivial solution is linear dependent. Th 1.4.3

Theroem3.3.1: Let, Then is linear independent PF:

Example 4. (P.150)

Theroem3.3.2: Suppose Then PF:

How to determine linear independency For the Vector Space P n (P.151) Question: Determine the linear dependency of Sol:

How to determine linear independency For the Vector Space C (n-1) [a,b] (P.152) Let Suppose

Wronskian Definition: Let be functions in C (n-1) [a,b], and define thus, the function is called the Wronskian of

Theroem3.3.3: Let if are linear dependent on [a,b] Cor:

Example of Wronskian Is linear independent in Yes, Example 6. (P.153) Is linear independent in Yes, Example 8. (P.154)

Question: Does the converse of Th hold? Answer: No, a counterexample is given as follows Question: Is linear independent in and Why?

§3-4 Basis and Dimension Definition: Let be a basis for a vector space if Example: It is easy to show that

Theroem3.4.1: Suppose PF:

Cor: If are both bases for a vector, then PF:

Dimension Definition: Let be a vector space. If has a basis consisting of n vectors, we say that has dimension n.  { } is said to have dimension 0.  is said to be finite dimensional if finite set of vectors that spans ;otherwise we say is infinite-dimensional.

Example of Dimension Example

Theroem3.4.3: If, then linear independent PF:

Theroem3.4.4:

Standard Basis

§3-5 Chang of Basis 不同場合用不同座標系統有不同的方便性，如質點 運動適合用體座標 (body frame) 來描述，而飛彈攔 截適合用球面座標。 利用某些特定基底表示時，有時更易使系統特性彰 顯出來。 Question: 不同座標系統間如何轉換？

Definition: Let be a vector space and let be an ordered basis for. unique expression

Remark 1: Lemma 2: Every n-dimensional vector space is isomorphic to

Example 4 (P.168) Question:

Example 4 (cont.) Solution:

Example 4 (cont.) Solution:

Transition Matrix Definition: V is called the transition matrix from the ordered basis F to the standard basis. Remark 1: V -1 is the transition matrix from to F. Remark 2: S=V -1 W is the transition matrix from E to F. V -1 W V -1 W P.169 figure changing coordinates in R 2

Theorem (P.171)

Theorem (cont.) PF:

Theorem (cont.) Remark : Corr. :

Example 6 (P.170) Question:

Example 6 (cont.) Solution:

Example 6 (cont.) Solution:

Example 7 (P.172) Question: Solution:

Example 7 (cont.) Solution:

Application 1: Population Migration (P.164)

Application 1 (cont.)

Markov (P.165)  Application 1 is an example of a type of mathematical model called Markov Process.  The sequence of vectors is called a Markov Chain.  A is called stochastic matrices, which has special struc- ture in that its entries are nonnegative and its columns all add up to 1.  If A is n×n, then we will want to choose basis vectors so that the effect of the matrix A on each basis vector is simply to scale it by some factor λ j, that is,

§3-6 Row Space and Column Space Definition: Let Then,

Example 1 (P.175)

Theroem3.6.1: Two row equivalent matrices have the same row space. PF:

Rank Definition: The rank of a matrix A is the dimension of the row space of A. Remark 1: The nonzero row of the row echelon mat- rix will form a basis for the row space. Remark 2: To determine the rank of a matrix, we can reduce the matrix to the echelon matrix.

Example 2 (P.175)

Theroem3.6.2: is consistent PF: Consistency Theorem for Linear System

Theroem3.6.3: Let, then PF:

Corollary 3.6.4: Definition:

Theroem3.6.5: Let, then PF: The Rank-Nullity Theorem

Example 3 (P.177)

Remark

Theroem3.6.6: Let, then PF: