Introduction to Linear Algebra 1 Flashcards

Questions and Answers

What is a linear equation?

Any equation that cannot be solved

An equation involving only one variable

An equation that graphs as a parabola

An equation in the form: $a_1x_1+a_2x_2+...+a_nx_n=b$ (correct)

What is a system of linear equations?

A collection of linear equations that use the same set of variables.

What is a solution to a linear system?

An ordered n-tuple of real numbers that makes every equation true.

What does solving a linear system mean?

Finding the solution set, i.e., the set of all possible solutions. Signup and view all the answers

Two linear systems are equivalent if they have the same solution set.

True Signup and view all the answers

What is the result of solving a two-variable system?

Finding the intersection of two lines. Signup and view all the answers

A matrix is defined as:

A rectangular array of numbers aligned in rows and columns. Signup and view all the answers

What is the Gauss elimination method?

A method that involves keeping a term in one equation and eliminating it from others. Signup and view all the answers

Match the following terms with their definitions:

Replacement = Add a multiple of one equation to another Interchange = Swap 2 equations Scaling = Multiply all terms in one equation by a nonzero constant Elementary row operation = Row replacement, row interchange, row scaling Signup and view all the answers

What is a linear system's fundamental question?

Is the system consistent, if a solution exists, is it unique? Signup and view all the answers

What is a column vector?

A matrix with one column. Signup and view all the answers

What does it mean if a system is homogeneous?

The system is defined by $b=0$. Signup and view all the answers

The homogeneous system is always consistent.

True Signup and view all the answers

X=0 is the ______ solution.

trivial Signup and view all the answers

Describe the condition for a linear system to have a nontrivial solution.

The system has infinite many solutions and at least one free variable. Signup and view all the answers

V1...vp are linearly independent if there is a nontrivial linear relation between them.

False Signup and view all the answers

What characterizes a linear transformation T?

It respects vector addition and scalar multiplication. Signup and view all the answers

If you have more vectors than the number of entries in each of them, they are independent.

False Signup and view all the answers

What is a transformation T:R^n -> R^m called if it respects vector addition and scalar multiplication?

Linear transformation. Signup and view all the answers

Study Notes

Linear Equations

A linear equation can be expressed in the form: ( a_1x_1 + a_2x_2 + ... + a_nx_n = b ), where ( a_1, a_2, ..., a_n ) are real numbers.

Systems of Linear Equations

A linear system is a collection of linear equations that share the same variables.
A solution to a linear system is an ordered n-tuple of real numbers that satisfies each equation in the system.

Solving Linear Systems

Finding the solution set of a linear system involves determining all possible solutions.
Two linear systems are equivalent if they share the same solution set.

Intersection of Lines

Solving a two-variable system equates to finding the intersection point of two lines.
A linear system can have no solutions, exactly one solution, or infinitely many solutions.

Matrices

A matrix is a rectangular arrangement of numbers organized into rows and columns.
Elementary row operations include row replacement, row interchange, and row scaling.

Gauss Elimination and Row Operations

The Gauss elimination method simplifies systems by keeping one variable and eliminating it from others, progressing through equations systematically.
Row equivalent matrices can be transformed into one another via a series of row operations.

Echelon Forms

A matrix is in Echelon form if all nonzero rows are above rows of zeros and if leading entries move rightward down rows.
A matrix is in Reduced Echelon form (rref) if it fulfills all criteria of Echelon form plus additional conditions on leading entries.

Consistency and Solutions

A linear system is consistent if the last column of its augmented matrix does not form a pivot column.
A consistent linear system has a unique solution when free variables are absent.

Matrix Operations and Vector Spaces

A column vector is a matrix with a single column.
( \mathbb{R}^n ) represents all vectors in n dimensions with real coefficients.

Linear Combinations and Spanning

The span of vectors ( v_1, ..., v_p ) is the set of all possible linear combinations.
A vector equation and its corresponding augmented matrix are equivalent if the resulting system is consistent.

Homogeneous Systems

A system is homogeneous if ( b = 0 ) and always has the trivial solution ( x = 0 ).
Nontrivial solutions exist in a homogeneous system if there are infinite solutions or free variables present.

Linear Dependence and Independence

Vectors ( v_1, ..., v_p ) are linearly dependent if a nontrivial linear combination equates to zero.
Linear independence occurs when the only solution to the equation involving the vectors is the trivial solution.

Transformations

A transformation ( T:\mathbb{R}^n \rightarrow \mathbb{R}^m ) is linear if it satisfies vector addition and scalar multiplication properties.
The augmented matrix transformation is a linear transformation.

Onto and One-to-One Mappings

A transformation is onto if every possible output in ( \mathbb{R}^m ) can be achieved, meaning ( Ax = b ) is consistent.
A transformation is one-to-one if the corresponding homogeneous equation has only the trivial solution and ensures no free variables exist.

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Description

Test your knowledge with these flashcards covering the basics of Linear Algebra. Each card presents key concepts including linear equations and systems of linear equations. Perfect for students looking to reinforce their understanding of introductory linear algebra topics.