Questions and Answers
A linear system whose equations are all homogeneous must be consistent.
True
Multiplying a linear equation through by zero is an acceptable elementary row operation.
False
The linear system x - y = 3, 2x - 2y = k cannot have a unique solution, regardless of the value of k.
True
A single linear equation with two or more unknowns must always have infinitely many solutions.
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If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
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If each equation in a consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c.
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Elementary row operations permit one equation in a linear system to be subtracted from another.
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If a matrix is in reduced row echelon form, then it is also in row echelon form.
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If an elementary row operation is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.
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Every matrix has a unique row echelon form.
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A homogeneous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's has n - r free variables.
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All leading 1's in a matrix in row echelon form must occur in different columns.
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If every column of a matrix in row echelon form has a leading 1, then all entries that are not leading 1's are zero.
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If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution.
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If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions.
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If a linear system has more unknowns than equations, then it must have infinitely many solutions.
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The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B.
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A set containing a single vector is linearly independent.
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The set of vectors (v, kv) is linearly dependent for every scalar k.
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Every linearly dependent set contains the zero vector.
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If the set of vectors is linearly independent, then kv1, kv2, kv2 is also linearly independent for every nonzero scalar k.
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If v1, ..., vn are linearly dependent nonzero vectors, then at least one vector vk is a unique linear combination of v1, ..., v(k-1).
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The set of 2x2 matrices that contain exactly two 1's and two 0's is linearly independent in M22.
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The functions f1 and f2 are linearly dependent if there is a real number x such that k1f1(x) + k2f2(x) = 0 for some scalars k1 and k2.
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A set with exactly one vector is linearly independent if and only if that vector is not 0.
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A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other.
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A finite set that contains 0 is linearly dependent.
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A vector is a directed line segment (an arrow).
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A vector is an n-tuple of real numbers.
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A vector is any element of a vector space.
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There is a vector space consisting of exactly two distinct vectors.
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What does linearly independent mean in terms of possible solutions?
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What does linearly dependent mean in terms of possible solutions?
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What is the matrix equation?
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Study Notes
Linear Systems and Homogeneity
- A linear system with all homogeneous equations is consistent.
- The system ax - by = c and 2ax - 2by = k cannot have a unique solution for any k.
- A single linear equation with multiple unknowns generally results in infinitely many solutions.
Elementary Row Operations
- Multiplying a linear equation by zero is not an acceptable elementary row operation.
- Elementary row operations allow subtraction of one equation from another.
- If an equation in a consistent linear system is multiplied by a constant, the transformation does not yield the same solutions.
Matrix Forms
- A matrix in reduced row echelon form is also in row echelon form.
- An elementary row operation may not maintain row echelon form.
- Every matrix can have non-unique row echelon forms.
Homogeneous Systems
- A homogeneous linear system with n unknowns presents n-r free variables when reduced to echelon form with r leading 1's.
- A system with n equations and n leading 1's has only the trivial solution.
Solutions and Inconsistencies
- The presence of a zero row in an augmented matrix does not assure infinitely many solutions.
- A system will not necessarily have infinitely many solutions even if it has more unknowns than equations.
Linear Dependence and Independence
- A set of vectors (v, kv) is always linearly dependent for any scalar k.
- Linear independence of a set of vectors becomes contingent on non-scalar multiples for two vectors.
- A set containing the zero vector is defined as linearly dependent.
Vector Spaces
- A vector is defined as any element belonging to a vector space.
- A vector must not be zero for a single vector to be considered linearly independent.
- A vector space cannot solely consist of two distinct vectors.
Solutions Implications
- Linear independence correlates with only the trivial solution existing.
- Linear dependence suggests the presence of a non-trivial solution.
Matrix Equation
- The fundamental matrix equation is represented as Ax = b.
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Description
This quiz covers essential concepts of linear algebra related to homogeneous systems, elementary row operations, and matrix forms. It explores consistency of linear equations, unique solutions, and the characteristics of matrices in reduced row echelon form. Test your understanding and enhance your knowledge in this fundamental area of mathematics.
