Questions and Answers
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.
True
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.
Signup and view all the answers
If the columns of an mxn matrix A span ℝ^m, then the equation Ax=b is consistent for each b in ℝ^m.
Signup and view all the answers
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.
Signup and view all the answers
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
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.
