Applet : Linear Independence/Dependence in R |
Let u and v be vectors in R^{3}. Their span is either a line through the origin or a plane through the origin in R^{3}. To check whether another vector WE Span {u, v} or not, geometrically it is whether w is along the line / plane separated by u, v. The applet below helps to analysis this. These vectors v_{1} , v_{2}, v_{3} in R^{3} are said to be linearly independent, if there exist scalars a, b, c (not all zero) such that a v_{1}+ b v_{2} + c v_{3 }= 0 Geometrically, this means that at least one of the vectors will lie in the plane generated by the other two. This applet helps you to view this. You can select the vectors either from the pre-set examples drop-down menu or click on the panel at any these points to generate v_{1} , v_{2}, v_{3}. Show button helps you to view the planes generated by two of them. You can rotate the plane by moving the cursor on the panel and see whether the third vector lie in the plane of the other two or not. |
In the above applet two examples are given. In which all the vectors of the example2 are in the same plane. You can see it by dragging the plane. In this applet you can choose different plane to see that all the vectors are in the same plane or not. |
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