Questions and Answers
Every elementary row operation is reversible.
True
A 5x6 matrix has 6 rows.
False
Two matrices are row equivalent if they have the same number of rows.
False
Elementary row operations used on an augmented matrix never change the solution set of the associated linear system.
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An inconsistent system has more than one solution.
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A consistent system of linear equations has one or more solutions.
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Two linear systems of equations are equivalent if they have the same solution set.
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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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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 [0 0 0 5 0], then the associated linear system is inconsistent.
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The echelon form of a matrix is unique.
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The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
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Reducing a matrix to echelon form is called the forward phase of the row reduction process.
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Whenever a linear system has free variables, the solution set contains many solutions.
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A general solution of a system is an explicit description of all solutions of the system.
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If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.
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An example of a linear combination of vectors v1 and v2 is the vector 1/2*v1.
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The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution of the equation x1a1 + x2a2 + x3a3 = b.
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The set Span{u,v} is always visualized as a plane through the origin.
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Any list of five real numbers is a vector in R5.
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When u and v are nonzero vectors, Span{u,v} contains the line through u and the origin.
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Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span{a1 a2 a3}.
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A vector b is a linear combination of the columns of a matrix A if and only if 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 m x n matrix A span Rm, then the equation Ax = b is consistent for each b in Rm.
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If A is an m x n 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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Every matrix equation Ax = b corresponds to a vector equation with the same solution set.
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Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.
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If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.
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If the augmented matrix [A b] has a pivot position in every row, then the equation Ax = b is inconsistent.
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If A is an m x n matrix whose columns do not span Rm, then the equation Ax = b is inconsistent for some b in Rm.
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A homogeneous equation 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 if and only if the equation has at least one free variable.
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The equation x = p + t v 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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If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero.
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The equation x = x2 u + x3 v with x2 and x3 free describes a plane through the origin.
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The equation Ax = b is homogeneous if the zero vector is a solution.
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The effect of adding p to a vector is to move the vector in a direction parallel to p.
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Assuming Ax = b has a solution, then the solution set of Ax = b is obtained by translating the solution set of 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 the other vectors in S.
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The columns of any 4 x 5 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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Two vectors are linearly dependent if and only if they lie on a line through the origin.
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If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
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If x and y are linearly independent, and if z is in Span{x,y}, then {x,y,z} is linearly dependent.
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If a set in Rn is linearly dependent, then the set contains more vectors than there are entries in each vector.
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If A is a 3 x 5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R3.
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If A is an m x n 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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Every matrix transformation is a linear transformation.
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The codomain of the transformation x->Ax is the set of all linear combinations of the columns of A.
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If T:Rn->Rm is a linear transformation and if c is in Rm, then a uniqueness question is "Is c in the range of T?"
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A linear transformation preserves the operations of vector addition and scalar multiplication.
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The superposition principle is a physical description of a linear transformation.
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A linear transformation T:Rn->Rm is completely determined by its effect on the columns of the n x n identity matrix.
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If T:R2->R2 rotates vectors about the origin through an angle x, then T is a linear transformation.
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When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
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A mapping T:Rn->Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.
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If A is a 3 x 2 matrix, then the transformation x -> Ax cannot be one-to-one.
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Not every linear transformation from Rn -> Rm is a matrix transformation.
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The columns of the standard matrix for a linear transformation from Rn -> Rm are the images of the columns of the n x n identity matrix.
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The standard matrix of a linear transformation from R2 to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form where a and d are +-1.
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A mapping T: Rn->Rm is one-to-one if each vector in Rn maps onto a unique vector in Rm.
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If A is a 3 x 2 matrix, then the transformation x->Ax cannot map R2 onto R3.
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Every matrix is row equivalent to a unique matrix in echelon form.
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Any system of n linear equations in n variables has at most n solutions.
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If a system of linear equations has two different solutions, it must have infinitely many solutions.
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If a system of linear equations has no free variables, then it has a unique solution.
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If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.
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If a system Ax = b has more than one solution, then so does the system Ax = 0.
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If A is an m x n matrix and the equation Ax = b is consistent for some b, then the columns of A span Rm.
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If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
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If matrices A and B are row equivalent, they must have the same reduced echelon form.
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The equation Ax = 0 has the trivial solution if and only if there are no free variables.
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If A is an m x n matrix and the equation Ax = b is consistent for every b in Rm, then A has m pivot columns.
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If an m x n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in Rm.
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If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.
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If 3 x 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
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Let A be an m x n matrix. If the equation Ax = b has at least two different solutions, and if the equation Ax=c is consistent, then the equation Ax=c has many solutions.
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If A and B are row equivalent m x n matrices and if the columns of A span Rm, then so do the columns of B.
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If none of the vectors in the set S= {v1, v2, v3} in R3 is a multiple of one of the other vectors, then S is linearly independent.
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If {u, v, w} is linearly independent, then u, v, and w are not in R2.
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In some cases, it is possible for four vectors to span R5.
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If u and v are in Rm, then -u is in span {u, v}.
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If u, v, and w are nonzero vectors in R2, then w is a linear combination of u and v.
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If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.
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Suppose that v1, v2, v3 are in R5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent.
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A linear transformation is a function.
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If A is a 6 x 5 matrix, the linear transformation x->Ax cannot map R5 onto R6.
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If A is an m x n matrix with m pivot columns, then the linear transformation x->Ax is a one-to-one mapping.
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Study Notes
Elementary Row Operations
- Every elementary row operation is reversible.
- Elementary row operations on an augmented matrix do not change the solution set of the corresponding linear system.
Matrix Dimensions
- A 5x6 matrix has 5 rows, not 6.
- Understand that the dimensions are given as rows x columns.
Row Equivalence
- Two matrices are row equivalent if they can be made identical through a series of elementary row operations.
- The uniqueness of reduced echelon form means two matrices cannot have different row reductions if they are equivalent.
Consistency of Systems
- A consistent system has one or more solutions, while an inconsistent system has no solutions.
- A linear system is inconsistent if there exists a row in an echelon form augmented matrix indicating a contradiction.
Variables and Solutions
- Basic variables correspond to pivot columns in the coefficient matrix, while free variables can lead to infinite solutions.
- A general solution encompasses all solutions of a system, expressed in parametric form.
Linear Combinations and Span
- A vector is a linear combination of others if it can be formed through vector addition and scalar multiplication.
- Span of a set of vectors describes the set of all possible linear combinations of those vectors.
Matrix Transformations and Linear Transformations
- Linear transformations preserve vector addition and scalar multiplication criteria, ensuring linearity.
- The standard matrix for a linear transformation reflects how the columns of the identity matrix map to Rm.
Independence and Dependence of Vectors
- A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of others.
- For linear independence, no vector can be written as a combination of the others in the set.
Solutions to Linear Equations
- If Ax = b is a system, it reflects the relationships among the columns of matrix A.
- The trivial solution of Ax = 0 exists for every homogeneous equation, with nontrivial solutions depending on free variables.
Transformations and Mappings
- A transformation x -> Ax is one-to-one if distinct inputs yield distinct outputs and onto if every output in Rm is covered.
- The effect of combining linear transformations preserves linearity, consistently mapping inputs to outputs.
Pivot Positions and Echelon Forms
- Each matrix equation corresponds to unique pivot positions that define its reduced echelon form.
- If an m x n matrix A has a pivot in every row, it suggests the related linear transformations have specific properties regarding uniqueness and span.
Linear Transformations in Higher Dimensions
- The ability of a matrix A (m x n) to span Rm depends on its rank and pivot structure.
- Transformations are defined in terms of both domain and codomain, with implications for consistency and solutions.
Miscellaneous Facts
- The superposition principle describes the linear combination behavior of transformations and physical systems.
- The uniqueness of solutions and their structures provide foundational insights into the nature of linear equations and transformations.
These notes encapsulate critical elements of linear algebra, emphasizing definitions, properties, and implications of matrices, transformations, and vector spaces.
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Test your understanding of linear algebra concepts with this True/False quiz. Each statement will challenge your knowledge of matrices and row operations. Assess your grasp of the course material effectively!
