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Quiz – 2

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Group 1
(One among the following set of questions)
1. (3 points) Consider the problem of computing the orthonormal basis of the subspace using the Grahm Schmidt Approach. Let W denote the subspace spanned by the vectors {v1,v2,v3}∈R3 where
     
1 0 c
      v1 =  a , v2 =  0  and, v3 =  0 
     
     
0 b 0 (1)
The first entry of the third orthonormal basis vector for the subspace W is (Please Note: Apply the Grahm Schmit algorithm by considering vectors in the following order, i.e., {v1,v2,v3})
2. (3 points) Consider the problem of computing the orthonormal basis of the subspace using the Grahm Schmidt Approach. Let W denote the subspace spanned by the vectors {v1,v2,v3}∈R3 where
     
1 0 c
      v1 =  a , v2 =  0  and, v3 =  0 
     
     
0 b 0 (4)
The second entry of the third orthonormal basis vector for the subspace W is (Please Note: Apply the Grahm
Schmit algorithm by considering vectors in the following order, i.e., {v1,v2,v3})
Group 2
(One among the following set of questions)
1. (3 points) Consider the problem of expressing the vector v ∈ R3 as a linear combination of combination of the
  x
 
basis vectors i.e., v . Let v =  y . Compute the coefficient of linear
 
  z
combination α3.
2. (3 points) Consider the problem of expressing the vector v ∈ R3 as a linear combination of combination of the
  x
 
basis vectorsi.e., v = P3i=1 αibi. Let v =  y . Compute the coefficient of linear
 
  z
 b1 b2 b3 
combination α1.
3. (3 points) Consider the problem of expressing the vector v ∈ R3 as a linear combination of combination of the
  x
 
basis vectors i.e., v . Let v =  y . Compute the coefficient of linear
 
  z
combination α2.
Group 3
(One among the following set of questions)
1. (1 point) For some a,b ∈Rn, that satisfy b ̸= αa, Projab = 0
A. True B. False
2. (1 point) For all a,b ∈Rn, that satisfy b ̸= αa, Projab = 0
A. True B. False
Group 4
(One among the following set of questions)
1. (1 point) Consider all possible sets of 3 linearly independent vectors (denoted by say, a,b,c ∈R3) such that b ⊥ a.
Let d = c−Projbc−Projac. Then, d ⊥ b and d ⊥ a.
A. True B. False
2. (1 point) Consider all possible sets of 3 linearly independent vectors (denoted by say, a,b,c ∈ R3) . Let d = c−Projbc−Projac. Then, d ⊥ b and d ⊥ a.
A. True B. False
Group 5
1. Consider all possible a,b,c ∈R3 such that a ⊥ b and b ⊥ c. Then a ⊥ c.
A. True B. False
Group 6
(One among the following set of questions)
  a
 
1. (3 points) The coordinate vector of x ∈R3 in basis. If xB =  b , the
 
  c
first entry of x in the standard basis is
  a
 
2. (3 points) The coordinate vector of x ∈R3 in basis. If xB =  b , the
 
  c
second entry of x in the standard basis is
  a
 
3. (3 points) The coordinate vector of x ∈R3 in basis. If xB =  b , the
 
  c
 b1 b2 b3 
third entry of x in the standard basis is