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: _________________
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Describe the elementary row operations.
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How are two matrices row equivalent?
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How do you know a solution set is infinite?
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When do elementary row operations stop?
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What is a matrix 'A' in echelon form?
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What makes matrix 'A' in RREF?
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Define pivot position.
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What is Theorem 1 regarding every matrix A?
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What is Theorem 2 about RREF of 'A'?
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What does a pivot column represent?
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What does a non-pivot column represent?
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A consistent linear system has many infinite solutions if and only if...
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What does vectors in R^2 represent?
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What is a scalar?
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Two operations in R^n are?
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Define span.
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What is a particular solution?
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If a1, a2, or a3 do not exist, is there a linear combination for 'b'?
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What defines a homogeneous linear system?
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What is the trivial solution?
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Does a homogeneous linear system have a non-trivial solution? Why?
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What characterizes a non-homogeneous linear system?
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What is the null space N(A)?
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What does it mean for A to be nonsingular?
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What does it mean for A to be singular?
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How do you write a solution set of Ax = 0?
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How do you write a solution set of Ax = b?
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What is Theorem 8 regarding linear dependence?
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If x and y are linearly independent, and (x, y, z) is linearly dependent, what can be said about z?
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If the columns of any 4 by 5 matrix are linearly independent, what can be inferred?
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What is a complete characterization of linearly dependent sets?
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How would you determine if a linear system has a nontrivial solution?
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If two vectors are multiples of each other, what is their linear relationship?
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If two vectors are not multiples of each other, what is their linear relationship?
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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
andy - 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.