8. Solving Ax = b: Row Reduced Form R

TL;DR

The rank of a matrix determines the number of solutions to a linear system.

Transcript

OK, when the camera says, we'll start. You want to give me a signal? OK, this is lecture eight in linear algebra, and this is the lecture where we completely solve linear equations. So Ax=b. That's our goal. If it has a solution. It certainly can happen that there is no solution. We have to identify that possibility by elimination. And then if ther... Read More

Key Insights

• 😜 The rank of a matrix is determined by the number of pivots in its row reduced echelon form.
• 😜 The rank can never exceed the number of columns.
• 😜 The rank determines the number of solutions to a linear system.
• 🔼 When the rank is smaller than the number of rows, there may be no solution or infinitely many solutions.

Questions & Answers

Q: What is the definition of matrix rank?

The rank of a matrix is the number of pivots present in its row reduced echelon form.

Q: Can a matrix have a rank greater than its number of columns?

No, the rank of a matrix can never exceed its number of columns.

Q: How does the rank affect the number of solutions to a linear system?

The rank determines the possible number of solutions: zero, one, or infinitely many.

Q: What is the difference between full column rank and full row rank?

Full column rank means that every column has a pivot, while full row rank means that every row has a pivot.

Q: Is there always a solution when the rank is smaller than the number of rows?

No, when the rank is smaller than the number of rows, there may be no solution or infinitely many solutions.

Q: What is the relationship between the null space and the rank?

The null space consists of vectors that, when multiplied by the matrix, yield zero. The dimension of the null space is equal to the number of free variables, which is the difference between the number of columns and the rank.

Q: How can one find the complete solution to a linear system?

The complete solution consists of a particular solution (found by setting all free variables to zero and solving for the pivot variables) and any vector in the null space multiplied by a constant.

Q: How does the rank affect the solvability of a linear system?

When the rank equals the number of columns, the linear system is solvable for any right-hand side. When the rank equals the number of rows, the linear system is solvable for every right-hand side.

Summary & Key Takeaways

• The rank of a matrix is the number of pivots in the row reduced echelon form.

• In the case of full column rank, there is either zero or one solution to the linear system.

• In the case of full row rank, there is always a solution, and it is either unique or infinitely many.

• When the rank is smaller than both the number of rows and columns, there is either no solution or infinitely many solutions.