Please neat handwriting. 1. Let x, y and z be vectors in a vector space V. Prove
ID: 3115414 • Letter: P
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Please neat handwriting.
1. Let x, y and z be vectors in a vector space V. Prove that if x+y = x+z then y = z, carefully stating all vector space axioms you use 2. Consider the set of all polynomials p(z) in Ps satisfying p(0)0. Is this a subspace of Ps? Justify your answer 3. Which of the following are spanning sets for R3? Justify your answers. 0a0 4. Which of the sets of vectors in Problem 3 are linearly independent in R3? Justify your answers. 5. a) What is the dimension of the span of each set of vectors in Problem 3? b) Find a vector that will extend 3a) to a basis of R3 c) Is 3b) a basis of R3? d) Which vector should be removed to make 3c) into a basis of R3? Justify your answers.Explanation / Answer
(b) The RREF of the 3x3 matrix with the given vectors as columns is
1
0
-2
0
1
2
0
0
0
Hence (2,2,0)T = -2(2,1,-2)T+2(3,2,-2)T. Thus, the given vectors are linearly dependent and hence they cannot constitute a basis for R3.
(c ) We know that the dimension of R3 is 3. Therefore, the given 4 vectors will be linearly dependent and hence they cannot constitute a basis for R3.
4. Only the set of vectors in Problem 3(a) is linearly independent in R3 as (0,1,2)T has 1 in the 2nd place so that (1,0,1)T cannot be a scalar multiple of (0,1,2)T and vice-versa.
5. (a) In 3(a) and 3(b), the dimension of the set is 2.In 3(b). In case of problem 3(c),the RREF of the 3x4 matrix with the given vectors as columns is
1
0
0
0
0
1
0
2
0
0
1
1
Hence only the first 3 vectors are linearly independent so that the dimension of the span of the set of given vectors is 3.
(b) The RREF of the 3x2 matrix with the given vectors as columns is
1
0
0
1
0
0
Hence the inclusion of the vector (0,0,1)T will make the set of vectors in 3(a), a basis for R3.
(c ) No, the set of vectors in 3(b) is linearly dependent and does not span R3. It cannot be a basis for R3.
(d) In view of the RREF of the 3x4 matrix with the given vectors as columns , the vector (1,2,3)T should be removed to make the set a basis for R3.
1
0
-2
0
1
2
0
0
0
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