Questions and Answers
A homogeneous equation is always consistent.
True
The equation Ax = 0 gives an explicit description of its solution set.
False
The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable.
False
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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If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero.
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The equation x = x2u + x3v, with x2 and x3 free (and neither u or v a multiple of the other), 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 the direction parallel to p.
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The solution set of Ax = b is obtained by translating the solution set of Ax = 0.
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The columns of the 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 × 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 the 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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A linear transformation is a special type of function.
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If A is a 3×5 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 × 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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A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
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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 columns of the n × n identity matrix.
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If T : R2 → R2 rotates vectors about the origin through an angle φ, then T is a linear transformation.
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When two linear transformations are performed one after another, then 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 3×2 matrix, then the transformation x → Ax cannot be one-to-one.
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Not every linear transformation from Rn to Rm is a matrix transformation.
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The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the n × 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 a 0 0 d, 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 3×2 matrix, then the transformation x → Ax cannot map R2 onto R3.
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Every elementary row operation is reversible.
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A 5×6 matrix has six rows.
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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.
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Two fundamental questions about a linear system involve existence and uniqueness.
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Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
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Two matrices are row equivalent if they have the same number of rows.
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An inconsistent system has more than one solution.
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Two linear systems 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 row 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 [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 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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The equation Ax = b is referred to as the vector equation.
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The 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×n matrix span Rm, then the equation Ax = b is consistent for each b in Rm.
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If A is an m×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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Study Notes
Homogeneous Equations
- A homogeneous equation always has a trivial solution, making it consistent.
- The equation Ax = 0 gives an implicit rather than an explicit description of its solution set.
- The trivial solution exists for Ax = 0 regardless of the presence of free variables.
Linearity and Vectors
- A line described by x = p + tv passes through point p and is parallel to vector v.
- The solution set of Ax = b takes the form w = p + vh if there exists a vector p such that Ap = b.
- Nontrivial solutions of Ax = 0 have at least one nonzero entry.
- The equation Ax = b is homogeneous if the zero vector is a solution, indicating b = 0.
Linear Independence and Dependence
- Linear independence of matrix A's columns is confirmed by Ax = 0 having only the trivial solution.
- A set is linearly dependent if at least one vector cannot be expressed as a combination of others; counterexamples exist.
- A 4 × 5 matrix has linearly dependent columns because it has more columns than rows.
- If vectors x and y are independent, and z is in Span{x, y}, the set {x, y, z} is dependent.
Transformations and Matrices
- Linear transformations are functions preserving vector addition and scalar multiplication properties.
- The transformation T(x) = Ax implies domain and codomain definitions, with corrections to misconceptions about dimensions.
- All matrix transformations are linear, but not all linear transformations are matrix transformations.
- A transformation is onto Rm only if its range matches the codomain.
Matrix Operations and Echelon Forms
- Each elementary row operation can be reversed, preserving the solution set of the associated linear system.
- Row equivalence is established through the ability to obtain one matrix from another through row operations.
- An inconsistent system implies no solutions, contrasting with equivalent systems having the same solution set.
Solution Sets
- A solution set can be described in the form of a list of numbers for a linear system, but this does not represent the entire solution set.
- The uniqueness of pivot positions depends solely on the row reduced echelon form, not on row interchanges.
- Presence of free variables in a system indicates multiple solutions.
General Solutions and Consistency
- A general solution provides an explicit description of all solutions within a system.
- The statement Ax = b describes a vector equation, subject to consistency checks based on the augmented matrix structure.
- Augmented matrices' pivot positions help determine the consistency of linear systems, along with the equations' conditions for spanning Rm.
Summary of Key Concepts
- The vector b can be expressed as a linear combination of A's columns if there is at least one solution to Ax = b.
- The forward phase of the row reduction process refers to transforming a matrix into echelon form.
- Statements regarding dimensions, mappings and matrix properties clarify misconceptions about linear algebra principles.
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Description
Explore the concepts of homogeneous equations, linear independence, and vector solutions in this quiz. Test your understanding of how these elements interact within linear algebra. This quiz covers fundamental principles that are essential for grasping more complex mathematical theories.
