14 11 this one is nearly definitional first note that

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§ 1.4 # 11 This one is nearly definitional. First, note that an arbitrary linear combination of elements of the set S = { x } is of the form v = a 1 x + a 2 x + · · · + a n x for some a 1 , . . . , a n F . By distributivity, v = ( a 1 + a 2 + · · · + a n ) x . By definition, span ( { x } ) is the set of all such linear combinations, which from our calculation is just { ax : a F } . Geometrically, in R 3 , this span is a line through the origin. § 1.5 # 1 (a) False. The set S = { (1 , 1) , (2 , 2) , (1 , 0) } ⊂ R 2 is linearly dependent (since the second vector is a non-zero multiple of the first), but the third vector is not a linear combination of the other two.
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HOMEWORK ASSIGNMENT 1 SOLUTIONS 5 (b) True. This is because 1 · ~ 0 = ~ 0, showing that there is a non-trivial linear combination equaling ~ 0. See the sentence on page 36, right before example 1. (c) False. See page 37, fact (1), following the definition of linear independence. (d) False. For example, the empty-set is not linearly dependent but it is always a subset. (e) True. This is the corollary on page 39. (f) True. By definition of linear independence. § 1.5 # 3 First notice that the problem has a typo. These are 3 × 2 matrices, not 2 × 3. By sight, we see that 1 1 0 0 0 0 + 0 0 1 1 0 0 + 0 0 0 0 1 1 + ( - 1) 1 0 1 0 1 0 + ( - 1) 0 1 0 1 0 1 = 0 0 0 0 0 0 . The general method is not much harder. Just think of 3 × 2 matrices as six-dimensional matrices. For example, think of the matrix 1 2 4 7 - 2 1 as the six-dimensional vector (1 , 2 , 4 , 7 , - 2 , 1). Then, to show that a set of matrices is linearly dependent, just think of them as row vectors, put them in a large matrix, and row reduce. This is Math 54 material. § 1.5 # 7 The set S = 1 0 0 0 , 0 0 0 1 is clearly linearly independent. Further, it consists of diagonal matrices in M 2 × 2 ( F ). Finally, any diagonal matrix a 0 0 b = a 1 0 0 0 + b 0 0 0 1 can be written as a linear combination of the elements of S , as above. Therefore, S is a basis for the diagonal matrices.
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