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[問題] 一條U yr1 MATHS VECTOR SPACE 的數 , help

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一條U yr1 MATHS VECTOR SPACE 的數 , help

Let  u = [1,2,3,-1,2]^T and v = [2,4,7,2,-1]^T in R^5 .
Find a basis of a space W such that w⊥u and w⊥v

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Notice that W = { (a, b, c, d, e) ∈ ℝ^5 : a + 2b + 3c − d + 2e = 0 and 2a + 4b + 7c + 2d − e = 0}. So a rough outline of how to tackle this problem is as follows:

(1) Determine a spanning set of vectors for W. This can be done by solving the following homogeneous system of  linear equations:
      a + 2b + 3c − d + 2e = 0
      2a + 4b + 7c + 2d − e = 0

(2) Then ensure that this set of vectors is a linearly independent set.
      (If you've been taught the Gram-Schmidt process, then this can be done "mechanically"  so that you can obtain an orthronormal basis for W. But I think here you don't need this.)

Hope this can help!

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