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Is resolution of forces similar to expressing a vector as a linear combination of some other vectors in linear algebra? Similarly, Is the state of stress at a point in 3D similar to expressing the applied load as a linear combination of 9 other vectors(the shear and normal components along the direction of the three axes)?

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Sort of.

When you resolve a vector into its component, you have to choose a coordinate system, resolving the vectors into components is the same as projecting the original vector on the coordinate axis. So yes the resolution is equivalent of linear combination.

If the stress tensor is symmetric then you need only six basis matrix to construct the linear space.

Otherwise you need nine component to describe the linear space. Notice, you cannot construct a general linear matrix space with vectors, you need matrixes. So the state of stress is generally the linear combination of nine other matrixes.

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    $\begingroup$ "if you are working in axisymmetric stress condition then you need three set of basis vectors." I know what you are trying to say, but that is only true if you use a cylindrical polar coordinate system aligned with the axis of symmetry. In any other coordinate system, you still have 6 non-zero stress components. The fact that the stress and strain distribution in the body does NOT depend on the coordinate system you choose to work in is important - especially if you want to progress from "following cookbook recipes" to "knowing how to cook" when doing continuum mechanics. $\endgroup$ – alephzero Oct 17 '18 at 8:38
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    $\begingroup$ @alephzero I removed that phrase, thanks for pointing it out. $\endgroup$ – Sam Farjamirad Oct 17 '18 at 8:57

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