True or False: - If W = span{x1,x2,x3} with {x1,x2,x3} linearly independent, and

Question

True or False:

- If W = span{x1,x2,x3} with {x1,x2,x3} linearly independent, and if {v1,v2,v3} is an orthogonal set of nonzero vectors in W, then {v1,v2,v3} is a basis for W.

- The Gram-Schmidt process produces from a linearly independent set {x1,...,xp} an orthogonal set {v1,...,vp} with the property that for each k, the vectors v1,...,vk span the same subspace as that spanned by x1,...,xk.

- If A=QR, where Q has orthonormal columns, then R=(Q^T)A

- If {v1,v2,v3} is an orthonormal basis for W, then multiplying v3 by a scalar c gives a new orthonormal basis {v1,v2,cv3}

- If x is not in a subspace W, then x-projw(x) is not zero.

(1 point) True False Problem a. If W Span {z! , z2,23} with {z! , z 2,23} linearly independent, and if {n, t 2, U3 } is an orthogonal set of nonzero vectors in W. then {n, v2, v3} is a basis for W. Choose b. The Gram-Schmidt process produces from a linearly independent set fxi,..., xp an orthogonal set fv,.., vp} with the property that for each k the vectors ,. ,vk Span the same subspace as that spanned by *i,.., Choose c. If A- QR, where Qhas orthonormal columns, then R-QTA. Choose d. If vi, 2, s is an orthonormal basis for W, then multiplying 's by a scalar c gives a new orthonormal basis svi, v2, cvs]. Choose e. If r is not in a subspace W, then r projw (x) is not zero. | Choose

Explanation / Answer

a.The statement is True. Since W = Span {x1,x2,x3} with {x1,x2,x3} linearly independent , hence {x1,x2,x3} is a basis for W and the dimension of W is 3. Further, since{v1,v2,v3} is a an orthogonal set of non-zero vectors in W, it is also a set of 3 linealy independent vectors in W. Hence {v1,v2,v3} is a basis for W.

b.The statement is True. The Gram-Schmidt process produces an orthogonal set of the same dimension as the original set of vectors. Further, an orthogonal set is linealy independent . If the dimensions of the original set and the orthogonal set derived by bthe Gram –Schmidt process are same, the 2 sets will span the same subspace.

c. The statement is True.

d. The statement is False. Each vector is an orthonormal basis is a unit vector. cv3 will not be a unit vector unless c = 1.

e. The statement is True. If x – projW (x )= 0, then x = projW (x ) W. Thus, if x is not in a subspace W, then x –projW(x) is not zero

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