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11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. %3D

11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. %3D

Advanced Engineering Mathematics
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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Please answer 11,12,13 this is my last ask
4. (AB)-1 = A-B-1 5. Suppose AB = AC and A is invertible. Then B = C. 6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R". 7. If the columns of an n x n matrix A span R", then the columns of A are linearly independent. 8. If the equation Ax = b has more than one solution for some b in R", then the columns of A span R". 9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. 10. Every square triangular matrix is invertible. 11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. 14. The dimension of Col(A) is the number of pivot columns of A. 15. Col(A) is the set of solutions to Ax = b. %3D
Transcribed Image Text:4. (AB)-1 = A-B-1 5. Suppose AB = AC and A is invertible. Then B = C. 6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R". 7. If the columns of an n x n matrix A span R", then the columns of A are linearly independent. 8. If the equation Ax = b has more than one solution for some b in R", then the columns of A span R". 9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. 10. Every square triangular matrix is invertible. 11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. 14. The dimension of Col(A) is the number of pivot columns of A. 15. Col(A) is the set of solutions to Ax = b. %3D
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