Let S = be a set of k vectors in R^, with k

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 21E
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Let S =
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 Rn.
Let A be an mxn matrix. Consider the statement. "For each b in Rm, the equation Ax = b has a solution." Because of a fundamental theorem about such matrix equations, this statement is equivalent to
what other statements? Choose all that apply.
A. The matrix A has a pivot position in each row.
B. Each bin Rm is a linear combination of the columns of A.
C. The matrix A has a pivot position in each column.
D. The rows of A span Rn.
E. The columns of A span Rm.
Transcribed Image Text:Let S = 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 Rn. Let A be an mxn matrix. Consider the statement. "For each b in Rm, the equation Ax = b has a solution." Because of a fundamental theorem about such matrix equations, this statement is equivalent to what other statements? Choose all that apply. A. The matrix A has a pivot position in each row. B. Each bin Rm is a linear combination of the columns of A. C. The matrix A has a pivot position in each column. D. The rows of A span Rn. E. The columns of A span Rm.
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