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# hw6 - Solutions of Selected Problems in HW6 3.4.8(a For...

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Solutions of Selected Problems in HW6 3.4.8 (a) For arbitrary ( a, b, c ) T R 3 , set C 1 1 1 1 + C 2 3 - 1 4 = a b c . The augmented matrix is 1 3 a 1 - 1 b 1 4 c of which RREF is 1 0 4 a - 3 c 0 1 - a + c 0 0 - 5 a + b + 4 c . There will be no solutions for C 1 , C 2 if - 5 a + b +4 c 6 = 0 . For example, ( a, b, c ) T = (0 , 1 , 1) T cannot be expressed as a linear combination of (1 , 1 , 1) T , (3 , - 1 , 4) T . Thus R 3 cannot be spanned by (1 , 1 , 1) T , (3 , - 1 , 4) T . (b) RREF( X ) should have no free variables so that the corresponding system for C 1 , C 2 , C 3 is consistent. Equivalently, you can also show det( X ) 6 = 0 . (Recall the theorem that for v 1 , · · · , v n R n , linear independence of the vectors are equivalent to their spanning R n . Thus it suffices to show either linear independence or spanning ability to show the vectors are a basis.) (c) From the part (a), you can set x 3 = ( a, b, c ) T with - 5 a + b + 4 c 6 = 0 . For example, x 3 = (0 , 1 , 1) T . Confirm det( X ) 6 = 0 .

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hw6 - Solutions of Selected Problems in HW6 3.4.8(a For...

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