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Questions and Answers
The equation Ax=b is referred to as a vector equation.
The equation Ax=b is referred to as a vector equation.
False
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.
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.
True
The equation Ax=b is consistent if the augmented matrix [ A b ] has a pivot position in every row.
The equation Ax=b is consistent if the augmented matrix [ A b ] has a pivot position in every row.
False
The first entry in the product Ax is a sum of products.
The first entry in the product Ax is a sum of products.
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If the columns of an mxn matrix A span ℝ^m, then the equation Ax=b is consistent for each b in ℝ^m.
If the columns of an mxn matrix A span ℝ^m, then the equation Ax=b is consistent for each b in ℝ^m.
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If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in ℝ^m, then A cannot have a pivot position in every row.
If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in ℝ^m, then A cannot have a pivot position in every row.
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Study Notes
Matrix Equation Overview
- The equation Ax=b is classified as a matrix equation, with A being a matrix.
- A vector b is a linear combination of the columns of A if and only if the equation Ax=b has at least one solution.
Consistency of the Equation
- The equation Ax=b is not necessarily consistent if the augmented matrix [ A b ] has a pivot position in every row; one pivot could be in the column for b.
- If the columns of an mxn matrix A span ℝ^m, Ax=b is consistent for every b in ℝ^m, ensuring a solution exists.
Pivots and Solutions
- Inconsistent equations indicate that A cannot have a pivot position in every row, meaning no solution exists for some b.
- The first entry in the product Ax represents a sum of products, calculated from the corresponding entries in x and the first column of A.
Studying That Suits You
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Description
Test your knowledge of the matrix equation Ax = b with these informative flashcards. Each card presents a true/false question that will help you understand the principles of linear combinations and matrix equations. Perfect for students looking to reinforce their understanding of linear algebra concepts.
