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In Algebra, "is" means Equal (=).
idiosyncrasies of matrix are the differences between matrix algebra and scalar one. i'll give a few examples. 1- in algebra AB=BA which sometimes doesn't hold in calculation of matrix. 2- if AB=0, scalar algebra says, either A, B or both A and B are equal to zero. this also doesn't hold in matrix algebra sometimes. 3- CD=CE taking that c isn't equal to 0, then D and # must be equal in scalar algebra. Matrix again tend to deviate from this identity. its to be noted that these deviations from scalar algebra arise due to calculations involving singular matrices.
Linear Algebra is a special "subset" of algebra in which they only take care of the very basic linear transformations. There are many many transformations in Algebra, linear algebra only concentrate on the linear ones. We say a transformation T: A --> B is linear over field F if T(a + b) = T(a) + T(b) and kT(a) = T(ka) where a, b is in A, k is in F, T(a) and T(b) is in B. A, B are two vector spaces.
It means "not equal to".
If a=b and b=c then a must = c