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3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x

3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.2: Guassian Elimination And Matrix Methods
Problem 93E
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Are the following statements true or false? ? ✓1. If the augmented matrix [A b] has a pivot position in every row, then the system Ax = b is inconsistent. 2. The equation Ax = b is referred to as a vector equation. ✓ 3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x ✓ 4. If the columns of an m x n matrix A span R™, then the equation Ax=b is consistent for each b in IR™ 5. If the system Ax = b is inconsistent, then b is not in the column space of A. ? 6. If A is an m x 7 matrix and if the equation Ax=b is inconsistent for some b in R™, then the RREF of A cannot have a pivot position in every row. 7. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution. ? ? ? ? ?
Transcribed Image Text:Are the following statements true or false? ? ✓1. If the augmented matrix [A b] has a pivot position in every row, then the system Ax = b is inconsistent. 2. The equation Ax = b is referred to as a vector equation. ✓ 3. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x ✓ 4. If the columns of an m x n matrix A span R™, then the equation Ax=b is consistent for each b in IR™ 5. If the system Ax = b is inconsistent, then b is not in the column space of A. ? 6. If A is an m x 7 matrix and if the equation Ax=b is inconsistent for some b in R™, then the RREF of A cannot have a pivot position in every row. 7. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution. ? ? ? ? ?
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