Matrix Equation Ax = b Flashcards

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Questions and Answers

The equation Ax=b is referred to as a vector equation.

False (B)

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 (A)

The equation Ax=b is consistent if the augmented matrix [ A b ] has a pivot position in every row.

False (B)

The first entry in the product Ax is a sum of products.

<p>True (A)</p> 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.

<p>True (A)</p> 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.

<p>True (A)</p> Signup and view all the answers

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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.

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