Questions and Answers
Every elementary row operation is reversible.
True
A 5 x 6 matrix has 6 rows.
False
The solution set of a linear system involving variables x1...xn is a list of numbers (s1...sn) that makes each equation in the system a true statement when the values s1...sn are substituted for x1...xn, respectively.
False
Two fundamental questions about a linear system involve existence and uniqueness.
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In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
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The row reduction algorithm applies only to augmented matrices for a linear system.
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A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
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Finding a parametric description of the solution set of a linear system is the same as solving the system.
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If one row in an echelon form of an augmented matrix is inconsistent, then the associated linear system is inconsistent.
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Another notation for the vector {-4, 3} is [-4, 3] (curly brace for vertical align).
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The points in the plane corresponding to {-2, 5} and {-5, 2} lie on a line through the origin.
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An example of a linear combination of vectors v1 and v2 is the vector 1/2v1.
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The solution set of a linear system whose augmented matrix is [a1, a2, a3, b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b.
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The set of Span {u, v} is always visualized as a plane through the origin.
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The equation Ax = b is referred to as a vector equation.
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A vector b is a linear combination of the columns of a matrix A iff the equation Ax=b has at least one solution.
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The equation Ax = b is consistent if the augmented matrix [A, b] has a pivot position in every row.
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The first entry in the product Ax is a sum of products.
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If the columns of an MxN matrix A span Rm, then the equation Ax = b is consistent for each b in Rm.
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If A is an MxN matrix and if the equation Ax = b is inconsistent for some b in Rm, then A cannot have a pivot position in every row.
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A homogeneous system is always consistent.
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The equation Ax = 0 gives an explicit description of its solution set.
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The homogeneous equation Ax = 0 has the trivial solution iff the equation has at least one free variable.
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The equation x = p + tv describes a line through v parallel to p.
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The solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the equation Ax = 0.
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The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution.
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If S is a linearly dependent set, then each vector is a linear combination of other vectors in S.
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The columns of any 4x5 matrix are linearly dependent.
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If x and y are linearly independent, and if [x, y, z] is linearly dependent, then z is in Span(x, y).
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A linear transformation is a special type of function.
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If A is a 3x5 matrix and T is a transformation defined T(x) = Ax, then the domain of T is R3.
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If A is an Mxn matrix, then the range of the transformation X -> Ax is Rm.
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Every linear transformation is a matrix transformation.
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A transformation T is linear iff (Tc1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T for all scalars c1 and c2.
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Study Notes
Elementary Row Operations
- Every elementary row operation is reversible, true by definition.
Matrix Dimensions
- A 5x6 matrix has 5 rows, not 6.
Solution Sets
- A solution set of a linear system is the collection of all possible solutions; it is not merely a list of numbers satisfying the equations.
Fundamental Questions
- Two key questions in linear systems are existence and uniqueness of solutions.
Reduced Echelon Form
- A matrix can only be row reduced to one unique reduced echelon form, regardless of the sequence of operations used.
Row Reduction Algorithm
- The row reduction algorithm applies to both augmented matrices and coefficient matrices.
Basic Variables
- A basic variable corresponds to a pivot column in the coefficient matrix of a linear system.
Parametric Descriptions
- Finding a parametric description of a solution set is equivalent to solving the linear system.
Echelon Form and Consistency
- A row in echelon form does not necessarily indicate inconsistency in the associated linear system.
Vector Notation
- The notation for a column vector uses parentheses (e.g., (-4, 3)), not curly braces.
Points in Plane
- The points corresponding to vectors {-2, 5} and {-5, 2} do not lie on a line through the origin; this can be verified by plotting.
Linear Combination
- A linear combination of vectors can take the form like 1/2v1, which includes scaling and summing vectors.
Linear System Representation
- The solution set of a linear system represented as [a1, a2, a3, b] corresponds to the linear combination x1a1 + x2a2 + x3a3 = b.
Span of Vectors
- The span of two vectors {u, v} is not always visualized as a plane; it degenerates to a line if v is a multiple of u.
Matrix Equation
- The equation Ax = b is classified as a matrix equation, not a vector equation.
Linear Combinations and Solutions
- A vector b is a linear combination of matrix A's columns if the equation Ax = b has at least one solution.
Pivot Positions
- An augmented matrix containing a pivot in every row does not guarantee consistency of the equation Ax = b.
Sum of Products
- The first entry in the product Ax is a sum of products of corresponding entries.
Consistency Conditions
- If the columns of a matrix A span Rm, then Ax = b is consistent for each b in Rm.
Pivot Positions and Inconsistency
- If Ax = b is inconsistent for some b in Rm, A cannot have pivot positions in every row.
Homogeneous Systems
- A homogeneous system (Ax = 0) is always consistent, potentially having multiple solutions.
Solution Descriptions
- The equation Ax = 0 does not offer an explicit description of its solution set; rather, it provides an implicit description.
Trivial Solutions
- The homogeneous equation Ax = 0 always has the trivial solution, regardless of the presence of free variables.
Line Equations
- The equation x = p + tv incorrectly describes a line; it should state that the line goes through p and is parallel to vector v.
Solution Sets and Vectors
- The solution set of Ax = b is not simply the set w = p + vh; it could be empty.
Linear Independence
- The definition of linear independence is not tied solely to the trivial solution in the context of the equation Ax = 0.
Linear Dependence
- A linearly dependent set does not imply every vector is a combination of others in the set.
Columns and Dependence
- In a 4x5 matrix, the columns are guaranteed to be linearly dependent due to the rule P < N (number of columns).
Vector Span and Dependency
- If x and y are linearly independent and [x, y, z] is dependent, then z must be within the Span{x, y}.
Linear Transformations
- A linear transformation is defined as a function with specific properties and can be represented as T(x) = Ax.
Transformation Domains
- If A is a 3x5 matrix and T(x) = Ax is defined, the domain of T is R^5, not R^3.
Image of Transformations
- The range of transformation X -> Ax is not R^m but is determined by the column space of A.
Types of Transformations
- Not every linear transformation is a matrix transformation; however, every matrix indeed represents a linear transformation.
Properties of Linear Transformations
- A transformation T is linear if it satisfies the condition T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1, v2 and scalars c1, c2 in its domain.
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Prepare for your Linear Algebra Test with these flashcards that cover essential concepts. Review the properties of row operations, matrix dimensions, and solution sets in linear systems. Ideal for students wanting to reinforce their understanding before the exam.
