of the following subspaces. (b) S2 = Span 2. Assume that fvi,...,n) is a basis f
ID: 3136396 • Letter: O
Question
of the following subspaces. (b) S2 = Span 2. Assume that fvi,...,n) is a basis for a vector space V. For some non-zero scalars (a) Prove that B spans V (b) Prove that B is a basis for V. , , 3. Let A E Mmxn (R). Let, , Uk,Uk+1, . . . ,Un} be a basis for Rn such that B is a basis for Null(A). Prove that C = {AUk+1, , A, is a basis for Col(A). 4. Consider the vector space S = {(a, b) R2 1 b > 0} with addition defined by (a, b)(c, d)-(ad+ bc, bd) and for any k E R scalar multiplication defined by k O (a, b) = (kat-, bk). (a) Prove that B = {(1,1), (1,2)) is a basis for S. (b) Find a basis for S that contains the vector (0, 2).Explanation / Answer
Let A =
It may be observed that the entries in the columns of A are the coefficients of E11,E12,E21,E22 in the vectors in S2.
The RREF of A is
Now, it is apparent that E11+5E12+E21+4E22 = 4(E11+2E12+E21+E22) -3(E11+5E12+E21+4E22) and -3E11-E12-3E21+2E22= 2(E11+2E12+E21+E22)-5(-3E11-E12-3E21+2E22) so that { E11+2E12+E21+E22, E11+E12+E21} is a basis for S2 i.e. the 1st 2 vectors in S2. The dimension of S2 is 2.
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