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Unformatted text preview: 1 X 1 + (1) X 2 + 0 X 3 + + 0 X m = 0 . 14. (a) Suppose that X m +1 is a linear combination of X 1 ,...,X m . Then h X 1 ,...,X m , X m +1 i = h X 1 ,...,X m i and hence dim h X 1 ,...,X m , X m +1 i = dim h X 1 ,...,X m i . (b) Suppose that X m +1 is not a linear combination of X 1 ,...,X m . If not all of X 1 ,...,X m are zero, there will be a subfamily X c 1 ,...,X c r which is a basis for h X 1 ,...,X m i . Then as X m +1 is not a linear combination of X c 1 ,...,X c r , it follows that X c 1 ,...,X c r , X m +1 are linearly independent. Also h X 1 ,...,X m , X m +1 i = h X c 1 ,...,X c r , X m +1 i . Consequently dim h X 1 ,...,X m , X m +1 i = r + 1 = dim h X 1 ,...,X m i + 1 . 37...
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 Fall '10
 Dreibelbis

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