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Question 04 (a)
Let W be the plane spanned by the vectors and . Verify that the vector is in the subspace W and show that the vectors u and w form an orthogonal basis for W.
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What is the relation between orthogonal vectors and linear independence?
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The vector u can be written as , so indeed .
To show that u and w form an orthogonal basis for W, we first check (or see by observation) that their inner-product is zero, , so they are orthogonal.
Now recall that in an inner-product space, orthogonal vectors are linearly independent, so u and w are linearly independent vectors in W.
Moreover, u and w span W, precisely because , so every vector in W is an element in .
Therefore u and w form a basis for W.