Prove these propositions. Proposition 1: has the following properties: (1) (x,y+z) - (x,y) + (x,z) (11) (x+y,z)

Prove these propositions.

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Proposition 1: has the following properties: (1) (x,y+z) - (x,y) + (x,z) (11) (x+y,z) - (x, z) + (y, z) (111) а(x, у) - (ах,уу , (x, ay) (x,y)a (iv) (x, y) (9,x) (v) (x,x) is always a real number 2 0 and (x,x) = 0 x (0,...,0) . (vi) If e…ºn is the standard basis for k" (e, - (0,...,0,1,0,...,0)) , T. then if 0 If 1+ j (vii) The inner product is nondegenerate; 1.e., If (x, y) = 0 for all then (0,...,0) ; X = If (x, y) = 0 for all y = (0,...,0). then Proposition 1: has the following properties: (1) (x,y+z) - (x,y) + (x,z) (11) (x+y,z) - (x, z) + (y, z) (111) а(x, у) - (ах,уу , (x, ay) (x,y)a (iv) (x, y) (9,x) (v) (x,x) is always a real number 2 0 and (x,x) = 0 x (0,...,0) . (vi) If e…ºn is the standard basis for k" (e, - (0,...,0,1,0,...,0)) , T. then if 0 If 1+ j (vii) The inner product is nondegenerate; 1.e., If (x, y) = 0 for all then (0,...,0) ; X = If (x, y) = 0 for all y = (0,...,0). then