Tools

Euclidean space as a vector space

Let RR3 = {(x,y,z)∣x,y,z∈R(x, y, z) | x, y, z ∈ R}. Then operations which can be performed in RR3 are
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Let A and B be the statements as given below :
A : RR4 = {(a,b,c,d)∣a,b,c,d∈R(a, b, c, d) | a, b, c, d ∈ R} is a vector space over RR
B : RR4 = {(a,b,c,d)∣a,b,c,d∈R(a, b, c, d) | a, b, c, d ∈ R} is a vector space over QQ
Then
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We can visualize geometrically the nn-dimensional Euclidean space, n∈Nn ∈ N,
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Let VV be a vector space over RR and let A and B be the statements as given below :
A : a+b=a+c⇒b=c;a+b = a+c ⇒ b=c; where a,b,c∈Va, b, c ∈ V
B : α.b=α.c⇒b=c;α.b = α.c ⇒ b=c; where α∈Fα ∈ F and b,c∈Vb, c ∈ V
Then
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