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up vote 6 down vote favorite 1 If you think of matrix-vector multiplication geometrically as a linear transformation to a new coordinate system and matrix-matrix multiplication as the composition of two separate linear transformations, what does it mean to add two matrices together? Would it make sense to think of it in terms of adding each basis vector separately to create a new set of basis vectors? linear-algebra matrices share | cite | improve this question edited Nov 20 at 18:22 asked Nov 20 at 18:17 ...

up vote 4 down vote favorite Let $T=S^1times S^1$ . There is a $mathbb Z_2$ -action on $T$ defined by $xsim -x$ (considering $T$ as a quotient of $mathbb R^2$ ). The quotient $X:=T/mathbb Z_2$ this has four singular points, say $x_1,...,x_4$ (corresponding to the four half lattice points in $mathbb R^2$ ). I want to calculate the fundamental group of $Xsetminus{x_1,...,x_4}$ . So far I could think of the following: Consider $Y:=Tsetminus{x_1,...,x_4}$ (abuse of notation). Then $Ytilde = S^1vee S^1vee S^1vee S^1vee S^1.$ Now $X=Y/mathbb Z_2$ . So we need to check how $mathbb Z_2$ acts on the $S^1$ -summands. From these we can choose three summands to be small circles around a singularity. The $mathbb Z_2$ -action on them is then simply the $mathbb Z_2$ -action on a circle with center $0$ in $mathbb R^2$ . So for them...