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Why would we want to transform a vector in our normal basis (xyz axes) to another basis? The only situation I can recall is when we are looking at a force applied on an inclined plane. Are there any other real life examples where this may be necessary?

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Computer graphics and perspective. When you view an object from two different points you are really having to change basis to describe the relative positions of the object. – fretty Nov 3 '12 at 10:21
Should be community wiki. Since you ask for many examples (and not a unique answer). – Julian Kuelshammer Nov 3 '12 at 10:21
up vote 2 down vote accepted

In any situation involving motion - a spaceship or a planet - there may be sets of global co-ordinates, but also local co-ordinates. For example, people used to measure the motion of the planets in a co-ordinate frame with the earth at the centre. Then they discovered that putting the sun at the centre made life simpler - orbits were conics with the sun at one focus. No-one has ever found an absolute frame of reference for the universe, so there is no privileged perspective - we can't arbitrate between different co-ordinate systems, so we have to be able to convert between them.

Even some simple questions of motion are made easier to analyse and understand in a frame of reference where the centre of mass is fixed.

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Let $A$ be an $n\times n$ matrix with eigenvalues $\lambda_1,\dots,\lambda_n$ and corresponding (linearly independent) eigenvectors $v_1,\dots,v_n$. Then for any (positive) integer $k$ and any vector $x$ we have $$A^kx=c_1\lambda_1^kv_1+\cdots+c_n\lambda_n^kv_n$$ where $c_1,\dots,c_n$ are the coordinates of $x$ with respect to the basis $\{{v_1,\dots,v_n\}}$. So if you find yourself in a situation where you are repeatedly multiplying by the matrix $A$ (and that is very common in applications), the calculations become very simple in the new basis.

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Whenever you whant to express a given vector as a sum of $n$ components in chousen $n-dimentional$ space. In the example you gave about forces on inclined plane, you try o express the force in $R^{2}$ that acts on the body ass composed of vertical and horizontal components respect to inclined plane of the displacement, but you could choose any other two, (not parallel) directions to express your force.

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