Linear Algebra Flashcards
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Questions and Answers

The size of the solution set is: ____________

infinite

How are two linear systems equivalent?

They share the same solution sets

How do you solve a linear system?

Replace the l.s. with one that is equivalent but simpler, then repeat until entries are cleared

A matrix with size 3x4 is: _________________

<p>augmented matrix</p> Signup and view all the answers

Describe the elementary row operations.

<p>Interchange, scaling, replacement</p> Signup and view all the answers

How are two matrices row equivalent?

<p>If one can be obtained from the other by a sequence of elementary row operations</p> Signup and view all the answers

How do you know a solution set is infinite?

<p>A free variable</p> Signup and view all the answers

When do elementary row operations stop?

<p>When you arrive at an equivalent matrix in reduced (row) echelon form (RREF)</p> Signup and view all the answers

What is a matrix 'A' in echelon form?

<p>All nonzero rows lie below the nonzero rows and all entries in a column below a leading entry are zeros</p> Signup and view all the answers

What makes matrix 'A' in RREF?

<p>All leading entries are '1', and every row starts with 1</p> Signup and view all the answers

Define pivot position.

<p>A position in matrix A corresponding to a leading entry in echelon form of A</p> Signup and view all the answers

What is Theorem 1 regarding every matrix A?

<p>Every matrix A row-reduces to RREF</p> Signup and view all the answers

What is Theorem 2 about RREF of 'A'?

<p>It is unique</p> Signup and view all the answers

What does a pivot column represent?

<p>A basic variable</p> Signup and view all the answers

What does a non-pivot column represent?

<p>A free variable</p> Signup and view all the answers

A consistent linear system has many infinite solutions if and only if...

<p>There is a free variable in a solution, i.e., r &lt; n</p> Signup and view all the answers

What does vectors in R^2 represent?

<p>v = (v1, v2)</p> Signup and view all the answers

What is a scalar?

<p>A (real) number used to multiply a vector or matrix</p> Signup and view all the answers

Two operations in R^n are?

<p>Add and subtract</p> Signup and view all the answers

Define span.

<p>Collection of all possible linear combinations</p> Signup and view all the answers

What is a particular solution?

<p>A solution set resulting from a parametric vector equation</p> Signup and view all the answers

If a1, a2, or a3 do not exist, is there a linear combination for 'b'?

<p>No</p> Signup and view all the answers

What defines a homogeneous linear system?

<p>A system that is of the form Ax = 0 vector in matrix equation form.</p> Signup and view all the answers

What is the trivial solution?

<p>The solution x-vector = 0 of a homogeneous equation Ax = 0</p> Signup and view all the answers

Does a homogeneous linear system have a non-trivial solution? Why?

<p>Yes, because it has a free variable</p> Signup and view all the answers

What characterizes a non-homogeneous linear system?

<p>Ax = b, b ≠ 0</p> Signup and view all the answers

What is the null space N(A)?

<p>The solution set of homogeneous linear system Ax = 0 vector</p> Signup and view all the answers

What does it mean for A to be nonsingular?

<p>N(A) = {0}</p> Signup and view all the answers

What does it mean for A to be singular?

<p>N(A) is infinite</p> Signup and view all the answers

How do you write a solution set of Ax = 0?

<p>N(A)</p> Signup and view all the answers

How do you write a solution set of Ax = b?

<p>p + N(A) where p is the particular solution of Ax = b</p> Signup and view all the answers

What is Theorem 8 regarding linear dependence?

<p>If there are more vectors than entries in the vector, the system is linearly dependent</p> Signup and view all the answers

If x and y are linearly independent, and (x, y, z) is linearly dependent, what can be said about z?

<p>z is in the span of {x, y}</p> Signup and view all the answers

If the columns of any 4 by 5 matrix are linearly independent, what can be inferred?

<p>Dependent</p> Signup and view all the answers

What is a complete characterization of linearly dependent sets?

<p>A set of vectors is linearly dependent if at least one of the vectors in the set can be expressed as a linear combination of the others.</p> Signup and view all the answers

How would you determine if a linear system has a nontrivial solution?

<p>Solve in matrix form and perform few row operations as possible</p> Signup and view all the answers

If two vectors are multiples of each other, what is their linear relationship?

<p>Dependent</p> Signup and view all the answers

If two vectors are not multiples of each other, what is their linear relationship?

<p>Independent</p> Signup and view all the answers

Study Notes

Solution Sets in Linear Algebra

  • A linear equation can have an infinite number of solutions if there are free variables present.
  • For example, the solution set of x + 2z = 4 and y - z = 0 is infinite.

Equivalent Linear Systems

  • Two linear systems are equivalent if they have identical solution sets.

Solving Linear Systems

  • To solve a linear system, replace it with an equivalent but simpler system, repeating until the entries are cleared.

Matrices

  • A matrix of size 3x4 is called an augmented matrix.
  • A matrix in echelon form has all nonzero rows above rows of all zeros.
  • A matrix is in reduced row echelon form (RREF) if leading entries are "1" and each leading entry is the only nonzero entry in its column.

Row Operations

  • Elementary row operations include:
    • Interchanging two rows.
    • Scaling a row by a non-zero constant.
    • Replacing one row by the sum of itself and a multiple of another row.

Row Equivalence

  • Two matrices are row equivalent if one can be transformed into another through a series of elementary row operations.

Echelon Forms

  • A matrix is in echelon form if all leading entries are positioned to the right of the leading entries in rows above.
  • The RREF has each leading entry as "1" and each leading column representing a pivot position.

Solutions of Linear Systems

  • A consistent linear system may have infinite solutions if it includes free variables (when the number of variables exceeds the number of equations).

Vector Spaces

  • Vectors in R^n have "n" components, where linear combinations can form planes, lines, or higher-dimensional spaces (e.g., R^2 represents a two-dimensional plane).
  • A scalar is a real number that multiplies a vector or matrix.

Linear Combinations and Span

  • The span of a set of vectors is the collection of all possible linear combinations, expressed in set notation.

Homogeneous Linear Systems

  • A system is homogeneous if it takes the form Ax = 0, which is always consistent with the trivial solution x = 0.
  • Homogeneous systems can have nontrivial solutions if free variables are present.

Linear Independence and Dependence

  • A set of vectors is linearly independent if the equation x1v1 + x2v2 + ... + xnvn = 0 has only the trivial solution, where all coefficients are zero.
  • Conversely, if it has a nontrivial solution, the vectors are dependent.

Matrix Properties

  • The null space of matrix A, denoted N(A), contains solutions of the homogeneous system.
  • A matrix A is nonsingular if N(A) = {0}, meaning it has a unique solution.
  • If A is singular, N(A) is infinite, indicating multiple solutions.

Pivot Columns and Free Variables

  • Pivot columns represent basic variables, while non-pivot columns represent free variables.
  • A consistent linear system shows an equal number of equations (m) and variables (n) when it has at least one free variable.

Applications and theorems

  • Various theorems highlight how linear combinations determine independence and dependence criteria, including relationships between vector sizes and matrix setup.
  • For instance, any set of vectors in R^n containing the zero vector is dependent, while more vectors than dimensions guarantee dependence.

General Concepts in Linear Algebra

  • Understanding concepts like span, linear independence, and the nature of solutions (trivial vs. non-trivial) is essential in mastering linear equations and matrices.
  • Working with geometrical interpretations helps visualize operations and relationships within vector spaces.

Strategies to Solve Linear Systems

  • Use row reduction techniques to simplify matrices and find general solutions in parametric vector form.
  • Seek to understand additional complexities introduced by free variables and their impact on solution sets.

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Test your understanding of key concepts in linear algebra with these flashcards. Cover essential topics such as solution sets, equivalence of linear systems, and methods for solving them. Perfect for students looking to solidify their knowledge.

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