5. Let H be the subspace of R4 of vectors that satisfy the equation 3r1-4r2-5r3+

Question

5. Let H be the subspace of R4 of vectors that satisfy the equation 3r1-4r2-5r3+6x4 0, and let K be the subspace of R4 of vectors that satisfy the symmetric equations 2:1 =12 = r3 = T4 (a) Find a basis for H (b) Find a basis for K (c) Extend your answer to (a) to a basis for R4 (d) Extend your answer to (b) to a basis for R (e) Verify that K is a subspace of H (f) Extend your answer to (b) to a basis for H 6. Suppose {vi, v2, va) is a basis for a vector space V. Determine whether each of the following are bases for V (a) fvi + 2v2 +3v3,2v1 +5v2 + 5v3, 2vi + 8v2 3v3] (b) (vi v2 - v3, V1-2v2v3) (c) (vi+ v2 +3v3,3vi + 2v2 + 7v3,2v - v2) (d) (vi+ v2, V1 + V3, V2+ v3, Vi - va) 7. Suppose A is a 5 x 7 matrix and rank A 3. (a) The column space of A is a subspace of Rn for n (b) What is the dimension of the column space of A? (c) The row space of A is a subspace of Rn for n = (d) What is the dimension of the row space of A? (e) The null space of A is a subspace of Rn for n = (f) What is the dimension of the null space of A? (g) The null space of AT is a subspace of Rn for n= (h) What is the dimension of the null space of AT? 8. Let U= span/ 1 2 and W = span S where S might be any of the four sets below. Which of those four choices of S would make RUW7 For those choices of S for which R- U+W, find ueU and we W such that |1 -u+vw (a)3 (b)10 13 (c) -2 (d) 11 9. Suppose that A is an m x n matrix. Prove that rank AT rank A.

Explanation / Answer

a) 3x1- 4x2-5x3+6x4 = 0

[1,1,1,1], [-2,0,0,1], [5,0,3,0] form a basis for H

b)K = span([1,1,1,1])

c)

Add a vector not in the subspace say [1,0,0,0]

d) Add vectors  [-2,0,0,1], [5,0,3,0],[1,0,0,0]

e) Since [1,1,1,1] is a basis element for H

f) Add vectors  [-2,0,0,1], [5,0,3,0],

2)

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