Let S= (v1. Vk) be a set of k vectors in R", with k
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let S= {v1.., vg} be a set of k vectors in R", with k<n. Use a theorem about the matrix equation Ax = b to explain why S cannot be a basis for R".
equations, this statement is equivalent to what other statements? Choose all that apply.
A. The matrix A has a pivot position in each column.
B. Each b in Rm is a linear combination of the columns of A.
C. The columns of A span Rm.
D. The matrix A has a pivot position in each row.
E. The rows of A span R".
Let A be the nxk matrix [v, v2 .-- Vk] whose columns are the vectors of S. Since A has fewer
of A.
than
there cannot be a pivot position in each
Therefore, by the above theorem, the
of A do not span
Why is the proof complete?
O A. The proof is complete because the subspace spanned by S is RK.
B. The proof is complete by the Spanning Set Theorem.
C. The proof is complete because a basis for a subspace must consist of linearly independent vectors.
O D. The proof is complete because a basis for a subspace must span that subspace.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90a00738-b6a4-4312-8152-8e6bd1a706fd%2F0e7b93cc-f0cc-42b3-a1ac-b19d1ee46f85%2Fnk81mli_processed.png&w=3840&q=75)
Transcribed Image Text:Let S= {v1.., vg} be a set of k vectors in R", with k<n. Use a theorem about the matrix equation Ax = b to explain why S cannot be a basis for R".
equations, this statement is equivalent to what other statements? Choose all that apply.
A. The matrix A has a pivot position in each column.
B. Each b in Rm is a linear combination of the columns of A.
C. The columns of A span Rm.
D. The matrix A has a pivot position in each row.
E. The rows of A span R".
Let A be the nxk matrix [v, v2 .-- Vk] whose columns are the vectors of S. Since A has fewer
of A.
than
there cannot be a pivot position in each
Therefore, by the above theorem, the
of A do not span
Why is the proof complete?
O A. The proof is complete because the subspace spanned by S is RK.
B. The proof is complete by the Spanning Set Theorem.
C. The proof is complete because a basis for a subspace must consist of linearly independent vectors.
O D. The proof is complete because a basis for a subspace must span that subspace.
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