vector mechanics

Aug 16th, 2016
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13Lecture 11 and 12(39) Suppose W1 , W2 are subspaces of a vector space V over F. Then defineW1 + W2 := {w1 + w2 : w1 W1 , w2 W2 }.This is a subspace of V and it is call the sum of W1 and W2 . Students must verify that W1 +W2is a subspace of V (use the criterion for a subspace).Examples:(a) Let V = R2 , W1 = {(x, x) : x R} and W2 = {(x, x) : x R}. Then W1 + W2 = R2 .Indeed, (x, y) = ( x+y, x+y) + ( xy, xy).22224(b) Next, let V = R , W1 = {(x, y, z, w) : x + y + z = 0, x + 2y z = 0}, W2 = {(s, 2s, 3s, t) :s, t R}. How to describe W1 + W2 (e.g., find a basis)?The following theorem tells us the dimension of W1 + W2 and the proof of the theorem suggesthow to write its bases.Theorem: If W1 , W2 are subspaces of a vector space V , thendim(W1 + W2 ) = dimW1 + dimW2 dim(W1 W2 ).Proof: Let S be a basis of W1 W2 (if W1 W2

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