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What is the physical significance of linearly independent and linearly dependent vectors
what is the physical significance of linearly independent and linearly dependent vectors
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Anshum Sharma![best-answer]()
Dependence in systems of linear equations means that two of the equations refer to the same line. There is an infinite number of solutions that will satisfy the conditions of the equations. To know which solution you want, you have to feed in an x value. This makes the y value dependent on the x value. Independence in systems of linear equations means that the two equations only meet at one point. There's only one point in the entire universe that will solve both equations at the same time; it's the intersection between the two lines. For example if you have a set of vector {x1,...,x5} and you can walk some distance in the x1 direction, then a difference distance in x2, then again in the direction of x3. If in the end you are back where you started then the vectors are linearly dependent (notice that I did not use all the vectors). This is the intuition behind the notion and you can make it into a definition because in the above example if we start at 0 then we walk ai in the xi direction, then the above paragraph says that a1x1+a2x2+a3x3=0. (This is how you should think of linear combinations, as directions to go given by your vectors.)