a. Let H = {(x_1, x_2)|x_1, x_2 greaterthanorequalto 0}. Is H a subspace of R^2?

Question

a. Let H = {(x_1, x_2)|x_1, x_2 greaterthanorequalto 0}. Is H a subspace of R^2? Why or why not? b. Let A be an n times n matrix and let b belongsto R^n. Consider the solution sets of Ax = 0 and Ax = b. Which solution set is not a subspace of R^n? Justify your answer. c. Let M_m times n be the set of all m times n matrices. It can be shown that M_m times n is a vector space under the usual operations of matrix addition and scalar multiplication. Let H be the set of all 2 times 2 real-valued matrices and let k belongsto R be a scalar. Show that H is a subspace of M m times n by showing that all three of the necessary properties are satisfied.

Explanation / Answer

5 a.

No.

(-2,-2) is in H,(1,3) is in H

Adding them gives

(-1,1) which i snot in H hence not a subspace

b.

Solution sets to Ax=b in general for b non zero

c.

If A,B are in H then A+B also has real entries ehnce is in H

If A is in H and c is a real number then cA also has real entries

HEnce, H is a subspace

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