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 and 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 1s has n-r free variables.
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All leading 1s 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 1s 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 1s, 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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If A and B are 2x2 matrices, then AB=BA.
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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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Tr(AB) = tr(A)tr(B)
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(AB)transpose=(A)trans(B)trans.
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Tr(A trans)=tr(A).
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Tr(cA)=ctr(A).
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If AC=BC, then A=B.
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If B has a column of zeros, then so does AB if this product is defined.
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If B has a column of vectors, then so does BA if this product is defined.
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A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order.
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(AB)trans=(B)trans(A)trans.
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The transpose of a product of any number of matrices is the product of the transposes in the reverse order.
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Two nxn matrices are inverses of one another iff AB=BA=I.
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A^2-B^2=(A-B)(A+B).
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(AB)^-1=A^-1B^-1.
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(kA+B)trans=k(A)trans + (B)trans.
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If A is an invertible matrix, then so is A trans.
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A square matrix containing a row or column of zeros cannot be invertible.
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The sum of two invertible matrices of the same size must be invertible.
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The product of two elementary matrices of the same size must be an elementary matrix.
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Every elementary matrix is invertible.
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If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent.
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If A is an nxn matrix that is not invertible, then the linear system Ax=0 has infinitely many solutions.
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If A is an nxn matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible.
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If A is invertible and a multiple of the first row of A is added to the second row, then the resulting matrix is invertible.
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An expression of the invertible matrix A as a product of elementary matrices is unique.
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The inverse of an invertible lower triangular matrix is an upper triangular matrix.
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A diagonal matrix is invertible iff all of its diagonal entries are positive.
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A matrix that is both symmetric and upper triangular must be a diagonal matrix.
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If A + B is symmetric, then A and B are symmetric.
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If A + B is upper triangular, then A and B are upper triangular.
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Det(A + B) = det(A) + det(B).
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Det(AB) = det(A)det(B).
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Det(A^{-1}) = 1/det(A).
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Every linearly independent subset of a vector space V is a basis for V.
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If v1, ..., vn is a basis for a vector space V, then every vector in V can be expressed as a linear combination of v1, ..., vn.
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The coordinate vector of a vector x in Rn relative to the standard basis for Rn is x.
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Every basis of P4 contains at least one polynomial of degree 3 or less.
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A vector space that cannot be spanned by finitely many vectors is said to be infinite-dimensional.
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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 so 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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The set of polynomials with degree exactly 1 is a vector space under the operations.
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Study Notes
Linear Systems and Solutions
- A homogeneous linear system is always consistent.
- Homogeneous systems can have infinitely many solutions, especially in under-determined cases.
- A linear system with dependent equations cannot have a unique solution.
Elementary Row Operations
- Multiplying an entire equation by zero is not a valid row operation.
- One equation can be subtracted from another using row operations.
- Elementary operations can change the form of matrices, but not their solution sets.
Matrix Forms
- Reduced row echelon form (RREF) includes all leading 1's in different columns.
- Row echelon form and RREF are related, where RREF is a stricter condition.
- Any matrix's row echelon form is not necessarily unique.
Inverses and Triangular Matrices
- A matrix with a row of zeros may have different implications for solutions, but does not automatically indicate infinitely many solutions.
- Non-invertible matrices can emerge if operations are performed that change their row structure.
- Invertible matrices maintain invertibility under certain operations.
Determinants and Matrix Operations
- The determinant of a product of matrices equals the product of their determinants.
- The determinant of the inverse matrix is the reciprocal of the original determinant; non-zero determinant indicates matrix invertibility.
Vector Spaces and Basis
- A basis for a vector space allows any vector in that space to be represented as a linear combination of basis vectors.
- Linear independence is defined by the inability to express any vector in the set as a combination of others.
- A finite set that includes the zero vector is always linearly dependent.
Linear Dependence and Independence
- A set of vectors is linearly dependent if one vector can be formed from the others.
- Every scalar multiple of a linearly independent vector maintains linear independence unless it's zero.
- A single vector is independent if it is not the zero vector.
Miscellaneous Properties
- A vector is defined as any element of a vector space, which may include varying dimensions.
- The set of all polynomials of a specific degree does not always form a vector space due to lack of closure under addition or scalar multiplication.
Studying That Suits You
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Description
Explore the intricacies of linear systems, elementary row operations, and matrix forms in this quiz. Test your understanding of homogeneous systems, RREF, and the properties of inverses and triangular matrices. Perfect for students looking to master linear algebra concepts.
