(7@) Which of the following sets of vectors span either R2 or R3, depending on w
ID: 2968976 • Letter: #
Question
(7@)
Which of the following sets of vectors span either R2 or R3, depending on which vector space the set belongs to?
A. [ 7 [ -5 [ 9
-6 3 -8
0] , 0] , 0]
____________________________-
B. [ 5 [6 [-1
2 -4 2
-7] , 2] , -5]
_________________________
C. [ 4 [ 4
-8] , -8]
________________________
D. [ 0 [2
0], - 6]
_______________________
E. [0 [2 [-1
0] , -6] , -5]
_____________________
F.
[7 [-5 [9
0 3 -8
6] , 2] , 0]
__________________
Please give the letters and a brief explanation why?
Explanation / Answer
for a set S which is subset of vector space V to span a vector space V,any element v of vector space V should be expressed as a linear combination of elements of S
A)
[ 7 [ -5 [ 9
-6 3 -8
0] , 0] , 0] cannot span R3 as z camponent is always zero for the linear combination of above elements
for (x,y) belonging to R2,there exists k1,k2,k3 such that
(x,y) = k1*(7,-6,0)+k2(-5,3,0)+k3*(9,-8,0)
therefore it spans R2
B. [ 5 [6 [-1
2 -4 2
-7] , 2] , -5] it spans R3 as the elements are linearly independent
since for (5,2,-7) = k1*(6,-4,2) + k2*(-1,2,-5) k1=k2=0
C. [ 4 [ 4
-8] , -8] is linearly dependent .it spans only R
D. [ 0 [2
0], - 6] is linearly dependent spans only R
E. [0 [2 [-1
0] , -6] , -5] spans R2 as any element (x,y) can be expressed as k1*(2,-6) + k2*(-1,-5)
F.
[7 [-5 [9
0 3 -8
6] , 2] , 0] is linearly independent and spans R3
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