1: Show that the set V = C = {x + yil x, y R} forms...

  

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1: Show that the set V = C = {x + yil x, y R} forms a vector space with usual addition and scalar multiplication of complex num- bers. = {(x, y) x, y 0} is not a vector Question 2: Show that the set V space with usual addition and usual scalar multiplication. Question 3: Consider the set V = R === {(x,y) x,y R} with usual addition and given scalar multiplication; a.(x, y) = (ax, ay - 1). (i). For u (-3, 1) and v = (4,-2) V. Find 1.u+ (-1).v. = (ii). Is V a vector space with usual addition and given scalar multiplication? If yes, prove it. If no, state at least one axiom that fails. Subspaces: Question 1: Write any two non-zero elements from the given subsets. (a). S ={(a, b, c)| a+b=2+c} CR. (b). S ={(a, b, c, d)| a+d=b+c} CR4. a b (c). S3 = { = c d |c=b and a d} C M2x2(R). Question 2: Decide which of the following subsets are subspaces of the given vector spaces. (a). W = {(a, b)| a+b=0} CR. (b). W = {(a, 0, c)| a, c ER} CR. = (c). W3 {a+ bx| a = b+1} C P(x). { [ (d). W = { (e). W = { W5 a b a c d b c d - |ad bc 0} M22 (R). |a+d=0} CM22(R). 1: Show that the set V = C = {x + yil x, y R} forms a vector space with usual addition and scalar multiplication of complex num- bers. = {(x, y) x, y 0} is not a vector Question 2: Show that the set V space with usual addition and usual scalar multiplication. Question 3: Consider the set V = R === {(x,y) x,y R} with usual addition and given scalar multiplication; a.(x, y) = (ax, ay - 1). (i). For u (-3, 1) and v = (4,-2) V. Find 1.u+ (-1).v. = (ii). Is V a vector space with usual addition and given scalar multiplication? If yes, prove it. If no, state at least one axiom that fails. Subspaces: Question 1: Write any two non-zero elements from the given subsets. (a). S ={(a, b, c)| a+b=2+c} CR. (b). S ={(a, b, c, d)| a+d=b+c} CR4. a b (c). S3 = { = c d |c=b and a d} C M2x2(R). Question 2: Decide which of the following subsets are subspaces of the given vector spaces. (a). W = {(a, b)| a+b=0} CR. (b). W = {(a, 0, c)| a, c ER} CR. = (c). W3 {a+ bx| a = b+1} C P(x). { [ (d). W = { (e). W = { W5 a b a c d b c d - |ad bc 0} M22 (R). |a+d=0} CM22(R).