Linear Algebra Flashcards

Questions and Answers

What are the 3 possible outcomes in a 2D linear system?

One unique solution

No Solution

Infinite Solutions

All of the above (correct)

What is the form of a general solution?

(x, y)

What are the steps to solving a linear system in 2D?

Multiply equations until same variable is each other's least common multiple. 2. Subtract equations to isolate a variable. 3. Plug isolated variable back into an original equation to find the other variable.

What do first degree equations refer to?

Linear systems Signup and view all the answers

How is a solution of an equation written?

In parentheses Signup and view all the answers

What is a consistent system?

A system with at least one solution Signup and view all the answers

What is an inconsistent system?

A system that has no solution Signup and view all the answers

What does 'the system is m equations in n unknowns' mean?

m = number of equations, n = number of variables Signup and view all the answers

What is an equivalent system?

Systems that have the exact same solution set Signup and view all the answers

What are the 3 ways of transforming systems?

<ol> <li>Exchange order of equations, 2) Multiply by a non-zero real number, 3) Add one equation to another</li> </ol> Signup and view all the answers

What does adding one equation to another look like?

Transforming the equations into a new equation Signup and view all the answers

What are the 7 steps of the Echelon Method?

<ol> <li>Ensure x term in first equation, 2) Eliminate x term in following equations, 3) Eliminate y term, 4) Eliminate z term, 5) Continue until Ay = K form, 6) Make first variable's coefficient 1, 7) Use substitution to find variables.</li> </ol> Signup and view all the answers

X1, x2, x3 are another way to say what?

x, y, z Signup and view all the answers

What is the Echelon Method with no solutions?

It doesn't produce a valid solution set. Signup and view all the answers

What is the Echelon Method with infinite solutions?

It indicates redundancy among equations. Signup and view all the answers

What is the difference between general and particular solutions?

General solution is in (x,y) form; particular solution is the general solution with substituted values. Signup and view all the answers

0 = 0 is a true statement.

True Signup and view all the answers

0 = 20 is a true statement.

False Signup and view all the answers

Which variable is the parameter in the equation x = y + 3, with a general solution of (y + 3, y)?

y Signup and view all the answers

The original equations in a system can be solved by both particular and general solutions.

True Signup and view all the answers

How can you make the solution logical?

If x and y represent the number of female and male workers, they cannot be negative. Signup and view all the answers

Why does the system have infinite solutions for x + y + z = 40 and y + 3z = 49?

There are 3 variables but only 2 equations. Signup and view all the answers

Study Notes

Outcomes in a 2D Linear System

Three possible outcomes: unique solution (lines intersect), no solution (parallel lines, inconsistent system), infinite solutions (lines are identical, equations are dependent).

General Solution

The general solution of a linear equation, expressed as (x, y), indicates all possible solutions when there is more than one unique solution.

Steps to Solve a 2D Linear System

Multiply equations to find the least common multiple for a variable.
Subtract equations to isolate one variable.
Substitute the isolated variable back to find the other variable.

Consistent and Inconsistent Systems

A consistent system has at least one solution (could be unique or infinite).
An inconsistent system has no solutions at all.

System of Equations

The notation "m equations in n unknowns" indicates m = number of equations, n = number of variables.

Equivalent Systems

Systems that share the exact same solution set; solutions must satisfy both systems to be considered equivalent.

Transforming Systems

Three transforming operations:
- Exchange the order of equations.
- Multiply an equation by a non-zero real number.
- Add one equation to another.

Echelon Method Steps

Confirm the presence of distinct variable terms in each equation.
Eliminate terms systematically from the first equation onward.
Formulate equations to simplify into the form Ay = K where A and K are constants.
Ensure the leading coefficient of the first variable is 1 for reduced echelon form.
Use substitution to solve each variable sequentially.

General vs. Particular Solutions

General Solution replaces one variable with an expression in (x, y) form.
Particular Solution arises from substituting a specific value into the general solution.

TRUE vs. FALSE Statements

Examples: 0 = 0 is true, while 0 = 20 is false.

Parameters in Solutions

In the equation x = y + 3, y acts as the parameter in its general solution (y + 3, y).

Existence of Solutions

A system may show infinite solutions when there are more variables than equations, indicating degrees of freedom.

Logical Solutions

When interpreting variables representing quantities (like workers), negative values are not applicable.

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Description

Test your understanding of key concepts in Linear Algebra with these flashcards. This quiz covers outcomes of linear systems and the general solution form in a 2D context. Enhance your skills through engaging terminology and definitions.