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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 18:38, 29 July 2023 Isaacreinhardt1 talk contribs created page 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...")
- 18:35, 29 July 2023 Isaacreinhardt1 talk contribs created page 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...")
- 22:30, 1 June 2023 Isaacreinhardt1 talk contribs created page 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...")
- 22:09, 1 June 2023 Isaacreinhardt1 talk contribs created page 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...")
- 21:56, 1 June 2023 Isaacreinhardt1 talk contribs created page 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 $\...")
- 21:03, 24 May 2023 Isaacreinhardt1 talk contribs created page Anticlockwise and Clockwise Plane Rotation Matrices are Inverses (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...")
- 22:53, 23 May 2023 Isaacreinhardt1 talk contribs created page Determinant of Anticlockwise 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?)
- 20:17, 23 May 2023 Isaacreinhardt1 talk contribs created page Matrix Equation of Plane Rotation/Clockwise (Created page with "== Theorem == <onlyinclude> Let $r_{-\alpha}$ be the clockwise 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' \en...")
- 20:05, 23 May 2023 Isaacreinhardt1 talk contribs created page Matrix Equation of Plane Rotation/Anticlockwise (I'm linking this to another page right now)
- 23:57, 22 May 2023 Isaacreinhardt1 talk contribs created page 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)")
- 22:25, 21 May 2023 Isaacreinhardt1 talk contribs created page 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...")
- 18:48, 21 May 2023 Isaacreinhardt1 talk contribs created page 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: :$\begin{bmatrix} x' \\ y' \end{bmatrix}$ = $\begin{bmatrix} \cos \alpha & -\sin \alph...")
- 01:56, 21 May 2023 Isaacreinhardt1 talk contribs created page Definition talk:Rotation Matrix (Created page with "== Convention? == I was wondering if anyone could describe what is intended to be on the page: === Convention === {{:Definition:Rotation Matrix/Convention}} On the definition page. I would like to work on a page about 2-D rotation matrices such as :$\mathbf R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and some results of it.")
- 02:19, 15 May 2023 Isaacreinhardt1 talk contribs created page Direction Angle of 2D Vector in Terms of Arctangent (I am new to ProofWiki, so I would appreciate feedback on its accuracy)
- 00:25, 12 May 2023 Isaacreinhardt1 talk contribs uploaded File:Cross product equivalency circle.png
- 00:25, 12 May 2023 Isaacreinhardt1 talk contribs created page File:Cross product equivalency circle.png
- 02:00, 11 May 2023 Isaacreinhardt1 talk contribs uploaded File:Cross Product Triangle.png
- 02:00, 11 May 2023 Isaacreinhardt1 talk contribs created page File:Cross Product Triangle.png
- 01:20, 11 May 2023 Isaacreinhardt1 talk contribs created page Talk:Equivalence of Definitions of Vector Cross Product (Work in progress:)