graphing-functionslinear algebrasystems of equations
I was trying to visualize how systems of linear equations in three unknowns work but failed miserably and I started to look for an online tool which would allow me to plot all three equations but surprisingly I could not find one.
Does anyone know of such precious tool?
One of the most frequent occasions where linear systems of $n$ equations in $n$ unknowns arise is in least-squares optimization problems. Let us look at an example. Let's say that we are studying two physical quantities $y$ and $x$ and we conjecture that $y$ is a second order polynomial function of $x$, i.e. $y=\alpha x^2 + \beta x + \gamma$ for some real numbers $\alpha$, $\beta$, $\gamma$ that are unknown. Let's say now that we perform experiments and obtain measurements $(x_1,y_1) \cdots (x_{100},y_{100})$. Applying the polynomial model on the measurements yields $y_i=\alpha x_i^2 + \beta x_i + \gamma$ for $i=1, \cdots 100$ or in matrix form $X k=y$ where $k=[\alpha \, \, \beta \, \, \gamma]^T$, $y=[y_1 \cdots y_{100}]^T$ and the $i^{th}$ row of $X$ is the row vector $[x_i^2 \, \, x_i \, \, 1]$. Now, as you might observe, we have $100$ equations in $3$ unknowns, i.e. our linear system $X k=y$ is overdetermined. Practically speaking, this system is consistent (i.e. it has a solution) only if indeed $y$ is related to $x$ via a second order polynomial equation (i.e. our conjecture is true) and additionally there is no noise in our measurements. So assume that none of the above two conditions is true. Hence the system $X k=y$ will not in general have a solution and one might consider finding a vector $k$ that instead minimizes $||X k - y||_2^2$, i.e. the square of the error. Then the solution of this optimization problem is the solution to the $3 \times 3$ system $X^T X k = X^T y$. This formulation comes up all the time in engineering, e.g. in signal prediction. So, least squares problems lead to square (i.e. $n \times n$) linear systems of equations.
A great site is GeoGebra. It starts in 2D by default, but you can click on a settings button on the right to open a 3D viewer. You can add a point anywhere on the page then double-click it to set its coördinates. There are many tools, including drawing the plane determined by three given points. One of the pleasures of this site is that you can drag any of the points and it will dynamically adjust the objects you have created (so dragging a point will move the corresponding plane).
Here is a screenshot of the plane through $(3,0,0),(0,2,0)$, and $(0,0,4)$:
$\hskip2cm$
Best Answer
Have you tried Geogebra 3D. It is not an online tool but it might be just what you need.geogebra
I know they also have an online version of the 2D version but I do not know if they have one of the 3D version yet. Please check the site.