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Given two sets S and T, the direct product of S and T is the...

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Basic Math

Given two sets S and T, the direct product of S and T is the setof ordered pairs S × T = {(s, t)|s ? S, t ? T}.Let V, W be twovector spaces over F.

(a) Prove that V × W is a vector space over F undercomponentwise addition and scalar multiplication (i.e. if (v1,w1),(v2, w2) ? V × W, then (v1, w1) + (v2, w2) = (v1+w1, v2+w2) anda(v, w) = (av, aw) for any (v, w) ? V ×W, a ? F).

(b) If dim V = n and dim W = m, prove that dim V × W = n + m byconstructing a basis.

Answer & Explanation Solved by verified expert
4.2 Ratings (604 Votes)
a Let V and W be 2 vector spaces over a field F and let v1 w1v2 w2 be 2 arbitrary elements of VxW and also let a be an arbitrary scalar Then v1 w1v2 w2 v1v2 w1w2 Furthersince V    See Answer
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