Let A be an mxn matrix. Explain why the equation Ax = b has a solution for all b in Rm if and only if the equation A'x =0 has only the trivial solution. Choose the correct answer below. O A. The system Ax = b has a solution for all b in R" if and only if the columns of A span Rm, or dim Col A = m. The equation A'x= 0 has only the trivial solution if and only if dim Nul A = 0. By the Rank Theorem, dim Col A = rank A = m - dim Nul A. Thus, dim Col A = m if and only if dim Nul A = 0. O B. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Col A =m. The equation A'x = 0 has only the trivial solution if and only if dim NulA' =0. Since Col A = Row A', dim Col A = dim Row A' = rank A' = m - dim Nul A' by the Rank Theorem. Thus, dim Col A = m Nul AT =0. and only if dim O C. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Row A =m. The equation A'x = 0 has only the trivial solution if and only if dim Nul A' = 0. Since Row A = Col A', dim Row A = dim Col A' = m - dim Nul A' by the Rank Theorem. Thus, dim Row A = m if and only if dim Nul A' = 0.
Let A be an mxn matrix. Explain why the equation Ax = b has a solution for all b in Rm if and only if the equation A'x =0 has only the trivial solution. Choose the correct answer below. O A. The system Ax = b has a solution for all b in R" if and only if the columns of A span Rm, or dim Col A = m. The equation A'x= 0 has only the trivial solution if and only if dim Nul A = 0. By the Rank Theorem, dim Col A = rank A = m - dim Nul A. Thus, dim Col A = m if and only if dim Nul A = 0. O B. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Col A =m. The equation A'x = 0 has only the trivial solution if and only if dim NulA' =0. Since Col A = Row A', dim Col A = dim Row A' = rank A' = m - dim Nul A' by the Rank Theorem. Thus, dim Col A = m Nul AT =0. and only if dim O C. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Row A =m. The equation A'x = 0 has only the trivial solution if and only if dim Nul A' = 0. Since Row A = Col A', dim Row A = dim Col A' = m - dim Nul A' by the Rank Theorem. Thus, dim Row A = m if and only if dim Nul A' = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Let A be an mxn matrix. Explain why the equation Ax = b has a solution for all b in Rm if and only if the equation A'x =0 has only the trivial solution.
Choose the correct answer below.
O A. The system Ax = b has a solution for all b in R" if and only if the columns of A span Rm, or dim Col A = m. The equation A'x= 0 has only the trivial solution if and
only if dim Nul A = 0. By the Rank Theorem, dim Col A = rank A = m - dim Nul A. Thus, dim Col A = m if and only if dim Nul A = 0.
O B. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Col A =m. The equation A'x = 0 has only the trivial solution if and
only if dim NulA' =0. Since Col A = Row A', dim Col A = dim Row A' = rank A' = m - dim Nul A' by the Rank Theorem. Thus, dim Col A = m
Nul AT =0.
and only if dim
O C. The system Ax = b has a solution for all b in R" if and only if the columns of A span R", or dim Row A =m. The equation A'x = 0 has only the trivial solution if
and only if dim Nul A' = 0. Since Row A = Col A', dim Row A = dim Col A' = m - dim Nul A' by the Rank Theorem. Thus, dim Row A = m if and only if dim Nul
A' = 0.
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