User contributions for Isaacreinhardt1
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24 November 2023
- 02:1002:10, 24 November 2023 diff hist 0 Tangents to Circle from Point subtend Equal Angles at Center/Corollary No edit summary
- 02:0902:09, 24 November 2023 diff hist +27 m Tangents to Circle from Point subtend Equal Angles at Center/Corollary No edit summary
2 September 2023
- 23:5923:59, 2 September 2023 diff hist +25 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 2 No edit summary
29 July 2023
- 18:5018:50, 29 July 2023 diff hist −86 User:Isaacreinhardt1 No edit summary current
- 18:4218:42, 29 July 2023 diff hist +148 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 2 No edit summary
- 18:4118:41, 29 July 2023 diff hist +148 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 1 No edit summary
- 18:4018:40, 29 July 2023 diff hist −683 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant No edit summary
- 18:3918:39, 29 July 2023 diff hist +28 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 1 No edit summary
- 18:3818:38, 29 July 2023 diff hist +966 N Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 1 Created page with "==Theorem== {{:Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant}} ==Proof== Let $\mathbf I_n$ denote the unit matrix of order $n$. {{begin-eqn}} {{eqn | l = \map \det {\mathbf A} \cdot \mathbf I_n | r = \mathbf A \cdot \adj {\mathbf A} | c = Matrix Product with Adjugate Matrix }} {{eqn | l = \map \det {\mathbf A} \cdot \mathbf A^{-1} \cdot \mathbf I_n | r..."
- 18:3718:37, 29 July 2023 diff hist +191 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant No edit summary
- 18:3718:37, 29 July 2023 diff hist +28 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 2 No edit summary
- 18:3518:35, 29 July 2023 diff hist +3,598 N Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 2 Created page with "==Theorem== {{:Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant}} ==Proof== Let $\mathbf A = \begin {bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end {bmatrix}$. Let $\mathbf A^{-1} = \begin {bmatrix} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nn} \end {bmatrix}$. Let $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}$ be the Definition:Standard Ordered..."
- 17:2017:20, 29 July 2023 diff hist +27 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant No edit summary
10 July 2023
- 19:3819:38, 10 July 2023 diff hist +845 Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant No edit summary
2 June 2023
- 18:0918:09, 2 June 2023 diff hist −5 m Direction Angle of 2D Vector in Terms of Arctangent Linked to vectors category
1 June 2023
- 22:4622:46, 1 June 2023 diff hist −234 User:Isaacreinhardt1 No edit summary
- 22:3122:31, 1 June 2023 diff hist +41 m Equations defining Plane Reflection/Matrix No edit summary
- 22:3022:30, 1 June 2023 diff hist +1,516 N Inverse of Plane Reflection Matrix Created page with "== Theorem == Let $\mathbf R$ be the matrix associated with a reflection in the plane. :$\mathbf R = \begin{bmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{bmatrix}$ Then its inverse matrix $\mathbf R^{-1}$ is itself. == Proof == Consider $\mathbf R \mathbf R$: {{begin-eqn}} {{eqn | l = \mathbf R \mathbf R | r = \begin{bmatr..."
- 22:1122:11, 1 June 2023 diff hist +39 Matrix Equation of Plane Rotation No edit summary
- 22:1022:10, 1 June 2023 diff hist +2 Equations defining Plane Reflection/Matrix No edit summary
- 22:1022:10, 1 June 2023 diff hist +60 m Equations defining Plane Reflection/Matrix No edit summary
- 22:0922:09, 1 June 2023 diff hist +763 N Determinant of Plane Reflection Matrix Created page with "== Theorem == The matrix associated with a reflection of the plane has a determinant of $-1$. == Proof == From Matrix Equation of Plane Reflection, we have: {{begin-eqn}} {{eqn | l = \begin{vmatrix} \cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{vmatrix} | r = -\map \cos {2\alpha} \map \cos {2..."
- 21:5721:57, 1 June 2023 diff hist +111 Equations defining Plane Reflection No edit summary current
- 21:5621:56, 1 June 2023 diff hist +1,745 N Equations defining Plane Reflection/Matrix Created page with "== Theorem == <onlyinclude> Let $\LL$ be a straight line through the origin $O$ of a cartesian plane. Let the angle between $\LL$ and the $x$-axis be $\alpha$. Let $\phi_\alpha$ denote the reflection in the plane whose axis is $\LL$. Let $\..."
- 21:5621:56, 1 June 2023 diff hist −14 m Equations defining Plane Rotation No edit summary current
25 May 2023
- 22:0722:07, 25 May 2023 diff hist +1,037 Inverse of Plane Rotation Matrix No edit summary
- 21:4721:47, 25 May 2023 diff hist −259 User:Isaacreinhardt1 No edit summary
- 21:4121:41, 25 May 2023 diff hist −429 Inverse of Plane Rotation Matrix No edit summary
- 20:5820:58, 25 May 2023 diff hist +1,139 Matrix Equation of Plane Rotation No edit summary
- 20:5120:51, 25 May 2023 diff hist −810 Determinant of Plane Rotation Matrix No edit summary
- 01:3301:33, 25 May 2023 diff hist +5 m Inverse of Plane Rotation Matrix No edit summary
- 01:2901:29, 25 May 2023 diff hist +172 User:Isaacreinhardt1 No edit summary
24 May 2023
- 21:0321:03, 24 May 2023 diff hist +2,528 N Inverse of Plane Rotation Matrix Created page with "== Theorem == Let $\mathbf R_1$ be the matrix associated with an anticlockwise rotation of the plane about the origin through an angle of $\alpha$. :$\mathbf {R_1} = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$ Let $\mathbf R_2$ be the matrix associated wit..."
- 20:1920:19, 24 May 2023 diff hist −182 User:Isaacreinhardt1 No edit summary
- 20:1420:14, 24 May 2023 diff hist +836 Determinant of Plane Rotation Matrix Rename the title of this page to Determinants of Plane Rotation Matrices
23 May 2023
- 23:2623:26, 23 May 2023 diff hist +515 m User:Isaacreinhardt1 No edit summary
- 22:5722:57, 23 May 2023 diff hist +72 Matrix Equation of Plane Rotation No edit summary
- 22:5622:56, 23 May 2023 diff hist +27 Determinant of Plane Rotation Matrix No edit summary
- 22:5322:53, 23 May 2023 diff hist +904 N Determinant of Plane Rotation Matrix The proof would be the EXACT same for the determinant of a clockwise plane rotation matrix. Could I combine both of them into the same page?
- 20:2220:22, 23 May 2023 diff hist +94 Equations defining Plane Rotation No edit summary
- 20:2020:20, 23 May 2023 diff hist +27 Matrix Equation of Plane Rotation No edit summary
- 20:1820:18, 23 May 2023 diff hist +114 Matrix Equation of Plane Rotation No edit summary
- 20:0620:06, 23 May 2023 diff hist −1,259 m Matrix Equation of Plane Rotation No edit summary
22 May 2023
- 23:5723:57, 22 May 2023 diff hist +303 N Definition talk:Rotation (Geometry)/Plane Created page with "== Clockwise? == This seems to only be defined for anticlockwise rotations. How would a clockwise rotation be described using this notation? Could I use this, for example? $\map {r_{-\alpha} } P$ --Isaacreinhardt1 (talk) 23:57, 22 May 2023 (UTC)"
- 22:3222:32, 22 May 2023 diff hist +23 m Matrix Equation of Plane Rotation No edit summary
21 May 2023
- 22:2522:25, 21 May 2023 diff hist +2,823 N Inverse of Invertible 2 x 2 Real Square Matrix Created page with "== Theorem == Let $\mathbf A$ be an invertible $2 \times 2$ real square matrix defined as: :$\mathbf A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ Then its inverse matrix $\mathbf A^{-1}$ is: :$\mathbf A^{-1} = \dfrac {1} {\map \det {\mathbf A}} \begin {pmatrix} d & -b \\ -c & a \end {pmatrix}$ == Proof == We construct $\begin {pmatrix} \mathbf A & \mathbf I \e..."
- 18:5118:51, 21 May 2023 diff hist +44 Matrix Equation of Plane Rotation No edit summary
- 18:5018:50, 21 May 2023 diff hist +4 m Matrix Equation of Plane Rotation No edit summary
- 18:4818:48, 21 May 2023 diff hist +1,503 N Matrix Equation of Plane Rotation Created page with "==Theorem== Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$. Let $r_\alpha$ rotate an arbitrary point in the plane $P = \tuple {x, y}$ onto $P' = \tuple {x', y'}$ Then: :$\begin{bmatrix} x' \\ y' \end{bmatrix}$ = $\begin{bmatrix} \cos \alpha & -\sin \alph..."
- 01:5601:56, 21 May 2023 diff hist +422 N Definition talk:Rotation Matrix Created page with "== Convention? == I was wondering if anyone could describe what is intended to be on the page: === Convention === {{:Definition:Rotation Matrix/Convention}} On the definition page. I would like to work on a page about 2-D rotation matrices such as :$\mathbf R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and some results of it."