Linear Algebra: Linear Systems and ERO Flashcards

Questions and Answers

What are linear equations?

Equation of form ax + by = c that defines a line in a plane.

What is a linear system?

Finite collection of linear equations in the variables x1, x2,..., xn.

What are the 3 possibilities regarding solutions to a system?

Exactly 1 solution

Infinitely many solutions

No solutions

All of the above (correct)

What are the 3 operations that preserve the solutions of a system?

Multiply an equation by a nonzero constant Signup and view all the answers

What is Row Echelon Form (REF)?

<ol> <li>Row has nonzero entry, 2. All rows consisting entirely of 0's are at the bottom, 3. Leading 1 of a row lies to the right of leading 1's in rows above it.</li> </ol> Signup and view all the answers

What is Reduced Row Echelon Form (RREF)?

<ol> <li>Row has nonzero entry, 2. All rows consisting entirely of 0's are at the bottom, 3. Leading 1 of a row lies to the right of leading 1's in rows above it, 4. Column has a leading 1 -> all other entries in that column are 0's.</li> </ol> Signup and view all the answers

What are 3 facts about REF/RREF?

REF has different valid answers Signup and view all the answers

What is Gaussian Elimination / Gauss-Jordan Elimination?

<ol> <li>Identify the leftmost column containing a nonzero entry, 2. Interchange 2 rows to set a nonzero entry at the top, 3. Create a leading 1 in the top row, 4. Use row operations to get 0's below the leading 1, 5. Working bottom to top, use row operations to set 0's above the leading 1's.</li> </ol> Signup and view all the answers

What are leading variables?

Variables corresponding to columns with leading 1's. Signup and view all the answers

What are free variables?

Variables that are not leading variables. Signup and view all the answers

What is a homogeneous system?

An augmented matrix that has all 0's in the last column. Signup and view all the answers

What is a trivial solution?

When all the solutions to all the unknowns are clearly 0. Signup and view all the answers

What is a matrix?

Rectangular array of real numbers. Signup and view all the answers

What is compact notation?

A = [aij] Signup and view all the answers

What is a column vector?

n * 1 matrix. Signup and view all the answers

What is a row vector?

1 * m matrix. Signup and view all the answers

What is a square matrix?

n * n matrix. Signup and view all the answers

What is the main diagonal of a matrix?

Entries A11, A22, A33,..., Anm of a square matrix. Signup and view all the answers

What are the basic matrix operations?

Addition Signup and view all the answers

What is a linear combination?

Expression of form x1A1 + x2A2 + ... + xkAk where A1,..., Ak are m * n matrices and x1,..., xk are scalars. Signup and view all the answers

What is the transpose of a matrix?

The transpose of the matrix would be n * m, where the 1st row becomes the 1st column, 2nd row becomes the 2nd column, etc. Signup and view all the answers

What is the trace of a matrix?

The addition of the main diagonal. Signup and view all the answers

Study Notes

Linear Equations

Linear equations take the form ( ax + by = c ), defining a line in a plane.
Example: ( x + 7y = 3 ) or ( 3x + 2y - z = 0 ) illustrates linear equations.
Non-linear terms (like powers) disrupt linearity.

Linear System

A linear system consists of a finite collection of linear equations involving variables ( x_1, x_2, \ldots, x_n ).
Examples include:
- ( x + 3y = 1 )
- ( 4x - y = 0 )
- ( 3x_1 + 2x_2 - x_3 = 4 )
- ( 4x_1 + x_2 - x_3 = 1 )
- ( x_1 + x_2 + x_3 = 0 )

Solutions to Systems

Three possibilities for solutions:
- One solution (consistent system)
- Infinitely many solutions (consistent system)
- No solutions (inconsistent system)

Elementary Row Operations (ERO)

Three operations that maintain system solutions:
- Multiply an equation by a non-zero constant.
- Interchange the order of two equations.
- Add a multiple of one equation to another.

Row Echelon Form (REF)

A matrix in REF has:
- The first non-zero entry in each row is 1 (leading 1).
- Rows of all zeroes are at the bottom.
- Leading 1's shift to the right in successive rows.

Reduced Row Echelon Form (RREF)

RREF has all properties of REF, plus:
- In each column with a leading 1, all other entries are zero.

Facts about REF/RREF

Every matrix can be transformed into REF or RREF using ERO.
REF is not unique but has unique pivot positions.
RREF is unique for a given matrix representation.

Gaussian Elimination/Gauss-Jordan Elimination

Steps to perform:
- Identify the leftmost column with a non-zero entry.
- Interchange rows to position a non-zero entry at the top.
- Create a leading 1 in the top row.
- Use row operations to eliminate entries below the leading 1, repeating on the submatrix.
- For RREF, perform elimination above leading 1's from bottom to top.

Leading Variables and Free Variables

Leading variables correspond to columns with leading 1's.
Free variables are those not tied to leading variables.

Homogeneous System

An augmented matrix with all zeroes in the last column.
Homogeneous systems are always consistent.

Trivial Solution

When all variable values are zero, resulting in the solution set being ((0,0,...,0)).

Matrix and Operations

Matrices are rectangular arrays of real numbers.
Compact notation is expressed as ( A = [a_{ij}] ).
A column vector is defined as an ( n \times 1 ) matrix.
A row vector is a ( 1 \times m ) matrix.
A square matrix has dimensions ( n \times n ).
The main diagonal consists of elements ( A_{11}, A_{22}, ... A_{nn} ).

Matrix Operations

Key operations include:
- Addition of matrices.
- Scalar multiplication of matrices.

Linear Combination

Defined as a form ( x_1 A_1 + x_2 A_2 +...+ x_k A_k ), where ( A_1,...,A_k ) are matrices and ( x_1,...,x_k ) are scalars.

Transpose of a Matrix

Transpose of an ( m \times n ) matrix results in an ( n \times m ) matrix.
Row elements transform to column elements.

Trace of a Matrix

The trace of a matrix is calculated by summing its main diagonal entries.

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Description

Explore the fundamentals of linear systems and operations like Echelon Row Operations (ERO), Row Echelon Form (REF), and Reduced Row Echelon Form (RREF). This quiz covers key concepts such as linear equations and systems. Perfect for enhancing your understanding of Gaussian and Gauss-Jordan elimination methods.