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Q2 and Q10 WW7 - r<=> v1.v2 = 0 So you should apply...

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Sheet1 Page 1 Q2 and Q10 Qu2: The skew-symmetric matrices form a vector space. Its dimension is n(n-1)/2 or (n2+ n)/2 If S is the subspace of M6 consisting of all upper triangular matrices, then dim S = 6 + 5 + 4 + 3 + 2 + 1 If S is the subspace of M8 consisting of all diagonal matrices, then dim S =8 If S is the subspace of M8 consisting of all matrices with trace 0, so dim S= n²-1 = 63 Q9 Remeber that an orthogonal basis have to satisfy : v1.v2 = 0 ( scaLAR PRODUCT) An orthoNormal Basis have to to satisfy w1.w2 = 0 and ||w1|| = ||w2|| = 1 !!!!! ( ABSOLUTE VALUE) Q10 It appears that there is some problem with this question that I can't figure out. However if you had this on a final, you should do this : We are asked for a orthogonal basis. What you should understand by this is that you should find 2 (in this case) vectors that a
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Unformatted text preview: r <=> v1.v2 = 0 So you should apply the normal technique, verify if there are orthogonal and if there are NOT: apply Gram Schmidt ( to make t h I know people who didnt have 2 orthogonal vectors and they got the question. So jsut remember what I just wrote. AGAIN 2 Othogonals Vectors HAVE to Satisfy v1.v2 = 0 !!!! If you find for instance a basis of 3 Vectors for the kernel and your are asked to find an Orthogonal Basis you will have to apply Gram Schmidt to Make them Orthogonal !!! ( And then I you wil have to divide each of the vector by there absolute value ==> v1/||v1||=w1...
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