Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W =R". The goal is to show that the subspace W is actually all of R". In general, when are two subspaces known to be the same space? O A. Two subspaces are the same when they are spanned by the same vectors. O B. Two subspaces are the same when they are subsets of the same space. O c. Two subspaces are the same when one subspace is a subset of the other subspace. O D. Two subspaces are the same when they have the same dimension. To start, assume that W is a subspace of R" spanned by n orthogonal nonzero vectors. Call these vectors V1. V2. V3. ... Vn- By assumption, W = Span (v1, V2. V3. .. Vn). Which of the following statements follows immediately from the assumption that the vectors v, V2. V3. Vn are orthogonal and nonzero? O A. (V1. V2. V3. Vn) is linearly independent. O B. (1. V2. V3. Vn) is linearly dependent. Oc. R"- Span (1, V2. V3. Vn) OD. At least one element of (v1, V2. V3. ... Vn) is zero. How does this show that W = R"? O A. W= (V. V2, V3. Vn) =R". O B. The column vectors v, V2. V3. .. Vn span R" if and only if (v. V2. V3. Vn) is linearly independent. OC. The column vectors v. V2. V3. V. span R"if and only if (v1. V2. V3. Vn) is linearly dependent. O D. The vectors v. V2, Va. Vn are in both W and R".
Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W =R". The goal is to show that the subspace W is actually all of R". In general, when are two subspaces known to be the same space? O A. Two subspaces are the same when they are spanned by the same vectors. O B. Two subspaces are the same when they are subsets of the same space. O c. Two subspaces are the same when one subspace is a subset of the other subspace. O D. Two subspaces are the same when they have the same dimension. To start, assume that W is a subspace of R" spanned by n orthogonal nonzero vectors. Call these vectors V1. V2. V3. ... Vn- By assumption, W = Span (v1, V2. V3. .. Vn). Which of the following statements follows immediately from the assumption that the vectors v, V2. V3. Vn are orthogonal and nonzero? O A. (V1. V2. V3. Vn) is linearly independent. O B. (1. V2. V3. Vn) is linearly dependent. Oc. R"- Span (1, V2. V3. Vn) OD. At least one element of (v1, V2. V3. ... Vn) is zero. How does this show that W = R"? O A. W= (V. V2, V3. Vn) =R". O B. The column vectors v, V2. V3. .. Vn span R" if and only if (v. V2. V3. Vn) is linearly independent. OC. The column vectors v. V2. V3. V. span R"if and only if (v1. V2. V3. Vn) is linearly dependent. O D. The vectors v. V2, Va. Vn are in both W and R".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why W= R.
....
The goal is to show that the subspace W is actually all of R". In general, when are two subspaces known to be the same space?
O A. Two subspaces are the same when they are spanned by the same vectors.
O B. Two subspaces are the same when they are subsets of the same space.
OC. Two subspaces are the same when one subspace is a subset of the other subspace.
O D. Two subspaces are the same when they have the same dimension.
To start, assume that W is a subspace of R" spanned by n orthogonal nonzero vectors. Call these vectors v,, v2, V3, .., vn. By assumption, W = Span {v1, V2, V3, ...,
v,3. Which of the following statements follows immediately from the assumption that the vectors v,, V2, V3, ., v, are orthogonal and nonzero?
O A. {V1, V2. v3, ..., Vn} is linearly independent.
O B. {V1, V2. V3, ..., Vn) is linearly dependent.
Oc. R" = Span (v1, V2, V3, .., Vn}
O D. At least one element of {v1, V2, V3, ..., vn is zero.
How does this show that W = R"?
O A. w= (v,, v2, V3, ..., Vn) = R".
O B. The column vectors v1, v2, V3, ., V, span R" if and only if {v1, V2, V3, .., V is linearly independent.
O C. The column vectors v, V2, V3, .., Vn span R" if and only if (v1, V2, V3, ..., Vn is linearly dependent.
O D. The vectors v1, V2, V3, ..., Vo are in both W and R".
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