1. Consider the vector space Max. Determine if the following sets are subspaces: (a) The set...

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1. Consider the vector space Max. Determine if the following sets are subspaces: (a) The set of all diagonal 3 x 3 matrices. (b) The set of all invertible 3 x 3 matrices. (c) The set of all singular 3 x 3 matrices. 2. Consider the vector space F(-00, 00) which is the space of all real valued functions with domain R. Determine if the following sets are subspaces: (a) The set of all functions such that f(0) = 0. (b) The set of all functions such that f(0) = 1. (c) The set of all function such that f(-2)=-f(r) (odd functions) (d) The set of functions that are solutions to y"- y = 0. 3. For the 2nd order differential equation ay'"+by+cy=k, for what value(s) of k is the solution space of this differential equation a subspace of (-∞, ∞). Why? 4. Let A be an n x n matrix and let A be some fixed real number. Show that the vectors € R" that satisfy the equation AA are a subspace of R". 1. Consider the vector space Max. Determine if the following sets are subspaces: (a) The set of all diagonal 3 x 3 matrices. (b) The set of all invertible 3 x 3 matrices. (c) The set of all singular 3 x 3 matrices. 2. Consider the vector space F(-00, 00) which is the space of all real valued functions with domain R. Determine if the following sets are subspaces: (a) The set of all functions such that f(0) = 0. (b) The set of all functions such that f(0) = 1. (c) The set of all function such that f(-2)=-f(r) (odd functions) (d) The set of functions that are solutions to y"- y = 0. 3. For the 2nd order differential equation ay'"+by+cy=k, for what value(s) of k is the solution space of this differential equation a subspace of (-∞, ∞). Why? 4. Let A be an n x n matrix and let A be some fixed real number. Show that the vectors € R" that satisfy the equation AA are a subspace of R".

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1 Vector Space Masa a The set of all diagonal 3 x 3 matrices To determine if this set is a subspace we need to check if it satisfies the three properties of a subspace 1 The set is closed under additi... View the full answer