Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.5_vector_spaces [2021/10/26 23:55] – abrilbook:chap2:2.5_vector_spaces [2024/11/10 20:52] (current) – jv
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 +[[forcestorques|2. Balancing Forces and Torques]]
 +  * [[  2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]]
 +  * [[  2.2 Sets| 2.2 Sets ]]
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * [[  2.4 Fields| 2.4 Fields ]]
 +  * ** 2.5 Vector spaces **
 +  * [[  2.6 Physics application balancing forces| 2.6.  Physics application: balancing forces]]
 +  * [[  2.7 The inner product | 2.7 The inner product]]
 +  * [[  2.8 Cartesian coordinates | 2.8 Cartesian coordinates]]
 +  * [[  2.9 Cross products --- torques| 2.9 Cross products — torques ]]
 +  * [[ 2.10 Worked example Calder's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 2.11 Problems| 2.11 Problems ]]
 +  * [[ 2.12 Further reading| 2.12 Further reading ]]
 +
 +----
 +
 ===== 2.5 Vector spaces ===== ===== 2.5 Vector spaces =====
  
 With the notions introduced in the preceding sections we can give now the formal definition of a vector space With the notions introduced in the preceding sections we can give now the formal definition of a vector space
-<WRAP box round>**Definition 2.9** <wrap em>Vector Space</wrap> \\+<WRAP box round #def_VectorSpace >**Definition 2.9** <wrap em>Vector Space</wrap> \\
 A //vector space// $(\mathsf{V}, \mathbb{F}, \oplus, \odot)$ is A //vector space// $(\mathsf{V}, \mathbb{F}, \oplus, \odot)$ is
 a set of //vectors// $\mathbf v \in \mathsf{V}$ a set of //vectors// $\mathbf v \in \mathsf{V}$
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 </WRAP> </WRAP>
  
-<WRAP box round #Matrix> **Definition 2.10** <wrap em>$N\times M$ Matrix: $\mathbb{M}^{N\times M}(\mathbb{F})$</wrap>\\+<WRAP box round #def_Matrix> **Definition 2.10** <wrap em>$N\times M$ Matrix: $\mathbb{M}^{N\times M}(\mathbb{F})$</wrap>\\
 For $N,M \in \mathbb{N}$ we define //$N\times M$ matrices// $A, B \in \mathbb{M}^{N\times M}(\mathbb{F})$ For $N,M \in \mathbb{N}$ we define //$N\times M$ matrices// $A, B \in \mathbb{M}^{N\times M}(\mathbb{F})$
 over the field $\mathbb{F}$ over the field $\mathbb{F}$
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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 2.16** <wrap em>$2\times 3$ matrices: summation and multiplication with a scalar</wrap> \\ +<WRAP box round>**Example 2.16** <wrap em>$3\times 2$ matrices: summation and multiplication with a scalar</wrap> \\ 
-To be specific we provide here the sum of two $2\times 3$ matrices and the multiplication by a factor of $\pi$.+To be specific we provide here the sum of two $3\times 2$ matrices and the multiplication by a factor of $\pi$.
 Let Let
 \begin{align*} \begin{align*}
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 </WRAP> </WRAP>
    
-<WRAP box round>**Example 2.17** <wrap em>Vector spaces: $M\times N$ matrices</wrap> \\+<WRAP box round #bsp_matrix-vector-space >**Example 2.17** <wrap em>Vector spaces: $M\times N$ matrices</wrap> \\
 The $N\times M$ matrices over a field $\mathbb{F}$,  $(\mathbb{M}^{N\times M}, \mathbb{F}, +, \cdot)$ The $N\times M$ matrices over a field $\mathbb{F}$,  $(\mathbb{M}^{N\times M}, \mathbb{F}, +, \cdot)$
 form a \\ form a \\
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 </WRAP> </WRAP>
  
-<WRAP box round #MatrixMultiplication >**Definition 2.11** <wrap em>Matrix multiplication</wrap> \\+<WRAP box round #def_MatrixMultiplication >**Definition 2.11** <wrap em>Matrix multiplication</wrap> \\
 For matrices one defines a product as follows For matrices one defines a product as follows
 \begin{align*} \begin{align*}
     \odot &: \mathbb{M}^{N\times L} \times \mathbb{M}^{L\times M}\to \mathbb{M}^{N\times M}     \odot &: \mathbb{M}^{N\times L} \times \mathbb{M}^{L\times M}\to \mathbb{M}^{N\times M}
     \\     \\
-    \forall A \in \mathbb{M}^{N\times L}, B \in \times \mathbb{M}^{L\times M} &:+    \forall A \in \mathbb{M}^{N\times L}, \: B \in \mathbb{M}^{L\times M} &:
     A \odot B = C = (c_{ij}) = \left( \sum_{k=1}^L a_{ik} b_{kj} \right)     A \odot B = C = (c_{ij}) = \left( \sum_{k=1}^L a_{ik} b_{kj} \right)
 \end{align*} \end{align*}
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 ==== 2.5.1 Self Test ==== ==== 2.5.1 Self Test ====
  
-<wrap #quest_vectorSelftest-checkProperties> Problem 2.11: ** Checking vector-space properties **\\+<WRAP #quest_vectorSelftest-checkProperties > Problem 2.11: ** Checking vector-space properties **
  
-  -  Verify that  $\mathbb{R}^D$ with the operations defined in [[#Example_VectorTuples |Example 2.15]] is a vector space.\\ +  - Verify that  $\mathbb{R}^D$ with the operations defined in [[#Example_VectorTuples |Example 2.15]] is a vector space. 
-  -  Verify that $N\times M$ matrices, as defined in [[#Matrix | Definition 2.10]], form a vector space. +  - Verify that $N\times M$ matrices, as defined in [[#def_Matrix|Definition 2.10]], form a vector space.
-</wrap>\\+
  
-<wrap #quest_groupSelftest-squareMatrixRotation >Problem 2.12: ** Geometric interpretation of matrices **</wrap>\\+</WRAP> 
 + 
 +----  
 + 
 +<WRAP #quest_groupSelftest-squareMatrixRotation >Problem 2.12: ** Geometric interpretation of matrices ** 
 +\\
 We explore the set of the eight matrices We explore the set of the eight matrices
 \begin{align*} \begin{align*}
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     \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix},     \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix},
     \begin{pmatrix} 0 & c \\ d & 0 \end{pmatrix},     \begin{pmatrix} 0 & c \\ d & 0 \end{pmatrix},
-                                 \quad\text{with \ } a, b, c, d \in \{ \pm 1 \}+                                 \quad\text{with } \quad a, b, c, d \in \{ \pm 1 \}
     \right\}     \right\}
 \end{align*} \end{align*}
  
-a) Let the action $\circ$ denotes matrix multiplication. Verify that $(M, \odot)$ is a group with respect to matrix multiplication, as defined in [[#MatrixMultiplication |Definition 2.11]]. We denote its neutral element as $\mathbb{I}$.\\+  * 1. Let the action $\circ$ denote matrix multiplication. Verify that $(M, \odot)$ is a group with respect to matrix multiplication, as defined in [[#def_MatrixMultiplication |Definition 2.11]]. We denote its neutral element as $\mathbb{I}$.
  
-b) Show that the group has five non-trivial elements $s_1, \dots s_5$ that are self inverse:+  * 2. Show that the group has five non-trivial elements $s_1, \dots s_5$ that are self inverse:
 \begin{align*} \begin{align*}
       s_i \neq \mathbb{I}       s_i \neq \mathbb{I}
       \quad \land \quad       \quad \land \quad
-      s_i \circ s_i = \mathbb{I} \quad \text{for \ \ } i \in \{ 1, \dots 6 \} \, .+      s_i \circ s_i = \mathbb{I} \quad \text{for } \quad i \in \{ 1, \dots 6 \} \, .
 \end{align*} \end{align*}
  
-c) Show that the other two elements $d$ and $r$ obey $d \circ r = r \circ d = \mathbb{I}$, +  * 3. Show that the other two elements $d$ and $r$ obey $d \circ r = r \circ d = \mathbb{I}$, \\ that $r = d \circ d \circ d$, and \\ that $d = r \circ r \circ r$.
-that $r = d \circ d \circ d$, and +
-that $d = r \circ r \circ r$.\\+
  
-d) Show that the set of points $P = \{ (1,1), (-1,1), (-1,-1), (1,-1) \}$ +  * 4. Show that the set of points $P = \{ (1,1), (-1,1), (-1,-1), (1,-1) \}$ is mapped to $P$ by the action of an element of the group:
-is mapped to $P$ by the action of an element of the group:+
 \begin{align*} \begin{align*}
       \forall m \in M \;\; \land \;\;  p \in P : \quad       \forall m \in M \;\; \land \;\;  p \in P : \quad
       p \circ m \in P       p \circ m \in P
 \end{align*} \end{align*}
-**Hint: ** The action of the matrix on the vector defined as follows+ 
 +++++ $\quad\quad\quad$Hint: | $\qquad\qquad\quad$ 
 +The action of the matrix on the vector is defined as follows
 \begin{align*} \begin{align*}
     (v_1, v_2) \circ \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22}  \end{pmatrix}     (v_1, v_2) \circ \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22}  \end{pmatrix}
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      \end{pmatrix}                                                       \end{pmatrix}                                                 
 \end{align*} \end{align*}
 +++++
  
-e) What is the geometric interpretation of the group $M$? +  * 5. What is the geometric interpretation of the group $M$? Illustrate the action of the group elements in terms of transformations of a suitably chosen geometric object. 
-Illustrate the action of the group elements in terms of transformations of a suitably chosen geometric object.+ 
 +</WRAP> 
 + 
 +---- 
 + 
 +<WRAP #quest_vectorSelftest-polynomials > Problem 2.13: ** Polynomials of degree $N$ **
  
-<wrap #quest_vectorSelftest-polynomials > Problem 2.13: ** Polynomials of degree $N$ **\\ 
 For a field $\mathbb{F}$ the polynomials $\mathsf{P}_N$ of degree $N$ in the variable $x$ are defined as For a field $\mathbb{F}$ the polynomials $\mathsf{P}_N$ of degree $N$ in the variable $x$ are defined as
 \begin{align*} \begin{align*}
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 \end{align*} \end{align*}
  
-a) State the rules of addition and multiplication with a scalar $s \in \mathbb{F}$ in analogy to the special case of $N=2$ discussed in [[#Example_VectorSpace-Polynomials |Example 2.18]].\\ +  - State the rules of addition and multiplication with a scalar $s \in \mathbb{F}$ in analogy to the special case of $N=2$ discussed in [[#Example_VectorSpace-Polynomials |Example 2.18]]. 
- +  - Verify that the polynomials of degree $N$ are a vector space. 
-b) Verify that the polynomials of degree $N$ are a vector space. +</WRAP>
-</wrap>+
  
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
book/chap2/2.5_vector_spaces.1635285347.txt.gz · Last modified: 2021/10/26 23:55 by abril