book:chap2:2.5_vector_spaces
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| book:chap2:2.5_vector_spaces [2021/10/27 00:17] – [2.5.1 Self Test] jv | book: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: | ||
| + | * [[ 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' | ||
| + | * [[ 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> | + | <WRAP box round # |
| A //vector space// $(\mathsf{V}, | A //vector space// $(\mathsf{V}, | ||
| a set of //vectors// $\mathbf v \in \mathsf{V}$ | a set of //vectors// $\mathbf v \in \mathsf{V}$ | ||
| Line 66: | Line 82: | ||
| </ | </ | ||
| - | <WRAP box round #Matrix> **Definition 2.10** <wrap em> | + | <WRAP box round #def_Matrix> **Definition 2.10** <wrap em> |
| 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 $\mathbb{F}$ | ||
| Line 86: | Line 102: | ||
| </ | </ | ||
| - | <WRAP box round> | + | <WRAP box round> |
| - | To be specific we provide here the sum of two $2\times | + | To be specific we provide here the sum of two $3\times |
| Let | Let | ||
| \begin{align*} | \begin{align*} | ||
| Line 128: | Line 144: | ||
| </ | </ | ||
| - | <WRAP box round> | + | <WRAP box round # |
| The $N\times M$ matrices over a field $\mathbb{F}$, | The $N\times M$ matrices over a field $\mathbb{F}$, | ||
| form a \\ | form a \\ | ||
| Line 135: | Line 151: | ||
| </ | </ | ||
| - | <WRAP box round #MatrixMultiplication | + | <WRAP box round #def_MatrixMultiplication |
| 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 | + | \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*} | ||
| Line 223: | Line 239: | ||
| - Verify that $\mathbb{R}^D$ with the operations defined in [[# | - Verify that $\mathbb{R}^D$ with the operations defined in [[# | ||
| - | - 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. |
| </ | </ | ||
| Line 240: | Line 256: | ||
| \end{align*} | \end{align*} | ||
| - | * 1. Let the action $\circ$ | + | * 1. Let the action $\circ$ |
| * 2. 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: | ||
| Line 251: | Line 267: | ||
| * 3. 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 $d \circ r = r \circ d = \mathbb{I}$, | ||
| - | * 4. 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 | ||
| Line 258: | Line 273: | ||
| \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 | ||
| \begin{align*} | \begin{align*} | ||
| (v_1, v_2) \circ \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} | (v_1, v_2) \circ \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} | ||
| Line 266: | Line 282: | ||
| | | ||
| \end{align*} | \end{align*} | ||
| + | ++++ | ||
| * 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. | * 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. | ||
book/chap2/2.5_vector_spaces.1635286668.txt.gz · Last modified: 2021/10/27 00:17 by jv