Consider the multiplicative matrix groups b GL(2,R) = {() = {(ad) | (a,b,c,d) R and ad...

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Consider the multiplicative matrix groups b GL(2,R) = {() = {(ad) | (a,b,c,d) R and ad be 40 - bc +0} and a b SL(2, R) = {()| (a,b,c,d) R and ad bc = 1 :=1}. (a) Show that GL(2, R) and SL (2, R) are topological groups. (b) Show that GL(2, R) is homeomorphic to an open subspace of R, while SL (2, R) is homeomorphic to a closed subspace of R. [Hint: Consider the determinant map Det : R R.] (c) Show that GL(2, R) is not connected, and SL (2, R) is noncom- pact. Consider the multiplicative matrix groups b GL(2,R) = {() = {(ad) | (a,b,c,d) R and ad be 40 - bc +0} and a b SL(2, R) = {()| (a,b,c,d) R and ad bc = 1 :=1}. (a) Show that GL(2, R) and SL (2, R) are topological groups. (b) Show that GL(2, R) is homeomorphic to an open subspace of R, while SL (2, R) is homeomorphic to a closed subspace of R. [Hint: Consider the determinant map Det : R R.] (c) Show that GL(2, R) is not connected, and SL (2, R) is noncom- pact.

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