Things to know - Chapters 1 & 2
MA2321, Fall '03, T. Olson

Know these terms and relationships between them:

• linear equation, system of linear equations
• equivalent systems of equations
• consistent/inconsistent
• coefficient matrix, augmented matrix
• elementary row operations
• echelon form, reduced echelon form
• pivot position, pivot column
• basic variable, free variable
• vector equation
• linear combination of vectors
• span of a set of vectors
• homogeneous equation, trivial solution
• parametric vector form (of general solution)
• linear independence/dependence
• linear transformation (2 properties)
• zero matrix, identity matrix
• matrix product
• inverse matrix, invertible matrix
• column space
• null space
• basis
• partition, block

Things to think about:

1.

What are the elementary row operations?

2.

How do you write a system of linear equations as a matrix equation? A vector equation?

3.

What are the two ways to compute the product of a matrix times a vector? ...a matrix times a matrix?

4.

When is matrix addition defined? ...matrix multiplication?

5.

What information can you get from the echelon form (or reduced echelon form) of a matrix? ...of an augmented matrix?

6.

How do you decide if a given vector is in the span of another set of vectors?

7.

How is the solution set of the homogeneous equation related to the solution set of the non-homogeneous equation?

8.

How do you tell if a set of vectors is linearly independent?

9.

How can you find the matrix of a linear transformation?

10.

Matrix algebra looks a lot like the algebra of scalar variables. Which property(ies) do NOT hold for matrix algebra? (e.g., distributive law, etc.) What does this mean for solving equations?

11.

How can you decide, using row reduction, if a matrix is invertible?

12.

Given an $3 \times 3$ matrix A, what statements can you write which are equivalent to saying ``The equation $Ax=\left(\begin{array}{r} 5 \\ -1 \\ 9 \end{array}\right)$ has exactly one solution''?

13.

For a square matrix, how is invertibility related to solving the homogeneous equation? ...non-homogeneous equations? ...the column space of the matrix? ...the null space of the matrix? ...the span of the columns?

14.

How can you tell if a given vector is in the column space of a matrix?

15.

How can you tell if a given vector is in the null space of a matrix?

16.

How do you find a basis for the column space of a matrix?

17.

How do you find a basis for the null space of a matrix?

18.

What is meant by the questions of ``existence and uniqueness'' for solutions to $A\vec{x}=\vec{b}$? How are these related to pivots in the echelon form of A?