Linear Algebra Exam #1 Flashcards

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Questions and Answers

What is a consistent system?

A system with fewer equations than variables

A system with no solution

A system with more equations than variables

A system with at least 1 solution (correct)

What is an inconsistent system?

A system with fewer equations than variables

A system with the exact same solution set as another system

A system with no solution (correct)

A system with at least 1 solution

What does equivalent refer to in the context of systems?

A system that has no solution

Two systems with the exact same solution set (correct)

Two systems with different solution sets

A system with fewer equations than variables

What is strict triangular form?

A form where the k-1 coefficients in the kth equation must be zero Signup and view all the answers

What does Row Echelon Form require?

First non-zero entry in each non-zero row is a 1 Signup and view all the answers

What is an underdetermined system?

A system with fewer equations than variables Signup and view all the answers

What does overdetermined mean?

A system with more equations than variables Signup and view all the answers

What is a lead variable?

A column with a leading 1 Signup and view all the answers

What defines a free variable?

A column without a leading 1 Signup and view all the answers

What is a homogeneous solution?

A solution where the right-hand side is all zeros Signup and view all the answers

What is the Consistency Theorem for Linear Systems?

The system Ax=b is consistent iff b can be written as a linear combination of the columns of A Signup and view all the answers

What is an identity matrix?

1 if i=j; 0 if i!=j Signup and view all the answers

What does nonsingular mean?

A matrix that can be inverted Signup and view all the answers

What is the multiplicative inverse?

A matrix B, when multiplied to A on either side, gives I, UNIQUE Signup and view all the answers

What are Type I Elementary Matrices?

Matrices formed by swapping row i and j of I Signup and view all the answers

What are Type II Elementary Matrices?

Matrices formed by multiplying a row of I by a nonzero vector Signup and view all the answers

What are Type III Elementary Matrices?

Matrices formed by adding multiple of one row to another row Signup and view all the answers

What does row equivalent mean?

There exists a sequence of elementary matrices when multiplied to one matrix gives you another Signup and view all the answers

An nxn matrix has a unique system if and only if?

A is nonsingular Signup and view all the answers

What defines an upper triangular matrix?

Aij=0 when i>j; non-zero entries are located in the upper triangle Signup and view all the answers

What defines a lower triangle matrix?

Aij=0 when i<j; only non-zero entries can be in the lower triangle Signup and view all the answers

Study Notes

Linear Algebra Concepts

A consistent system has at least one solution, indicating solvability.
An inconsistent system has no solution, making it unsolvable.
Equivalent systems share the same solution set, meaning they produce identical outcomes.
A strict triangular form in an ( n \times n ) system requires that for each ( k )-th equation, ( k-1 ) coefficients are zero, and the( k )-th coefficient is non-zero.
Row Echelon Form requires that the first non-zero entry in every non-zero row is a leading 1, and leading zeros increase in subsequent rows. All-zero rows are positioned below non-zero rows.
An underdetermined system has fewer equations than variables, allowing for multiple solutions.
An overdetermined system comprises more equations than variables and may lead to inconsistency.
A lead variable is associated with a column that contains a leading 1 in the matrix representation.
A free variable appears in columns that do not contain a leading 1, often leading to flexibility in solutions.
A homogeneous solution refers to a system where all entries on the right-hand side are zeros; such systems are always consistent.
A linear combination of vectors ( a_1, a_2, \ldots, a_n ) with scalars ( c_1, c_2, \ldots, c_n ) is expressed as ( c_1a_1 + c_2a_2 + \ldots + c_na_n ).
The Consistency Theorem for Linear Systems states that system ( Ax = b ) is consistent if ( b ) can be represented as a linear combination of the columns of matrix ( A ).
The identity matrix is defined as having 1s on the diagonal (when ( i = j )) and 0s elsewhere.
A matrix is termed nonsingular if there exists another matrix ( B ) such that ( AB = BA = I ), indicating invertibility.
The multiplicative inverse of a matrix ( A ) is matrix ( B ) that satisfies the equation ( AB = I ) and is unique.
Type I Elementary Matrices result from swapping two rows (i and j) of the identity matrix.
Type II Elementary Matrices are created by multiplying a row of the identity matrix by a non-zero scalar.
Type III Elementary Matrices are formed by adding a multiple of one row to another row within the identity matrix.
Two matrices are row equivalent if a sequence of elementary matrices can transform one into another.
An ( n \times n ) matrix has a unique system if and only if it is nonsingular.
An upper triangular matrix has entries ( A_{ij} = 0 ) for ( i > j ), placing non-zero entries solely in the upper triangle.
A lower triangular matrix contains entries ( A_{ij} = 0 ) for ( i < j ), allowing non-zero entries only in the lower triangle.

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Description

Prepare for your Linear Algebra Exam with these flashcards covering essential terms and definitions. Understand concepts like consistent systems, equivalent systems, and strict triangular form. Use these cards to reinforce your knowledge and succeed in Exam #1.