q3rs - is the zero vector which can never be part of a...

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Quiz III 1. Find two subspaces S and T of R 2 such that S T , the set vectors that are either in T or in S , is not a subspace. Then give an example of a linear combination of vectors from S T that is not in S T . Solution: Any pair of lines in R 2 will work as an example here. You could take S to be the set of scalar multiples of (1 , 1) and T to be the set of scalar multiples of (1 , - 2). Then (1 , 1) + (1 , - 2) = (2 , 1) is a linear combination of a vector in S and a vector in T , but it is not a scalar multiple of either (1 , 1) or (1 , - 2), so it is not in S T . 2. Find a basis for the image of the transformation T ( ⃗x ) = A⃗x , when A = 1 3 0 4 2 6 0 8 1 3 0 0 Solution : The second column is three times the ﬁrst column, and the third column
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Unformatted text preview: is the zero vector which can never be part of a linearly independent set of vectors. The rst and fourth columns are linearly independent because the fourth one has a zero in a row where the rst column does not. So, since the image is the set of linear combinations of the columns, { 1 2 1 , 4 8 } is a basis for it. That is the set that you would get by removing redundant vectors, but the second and fourth columns of A are also a basis for the image....
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This note was uploaded on 02/06/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.

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