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)?
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.