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Linear Algebra (True/False)
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Terms in this set (70)
A linear system whose equations are all homogeneous must be consistent.
True
Multiplying a linear equation through by zero is an acceptable elementary row op.
False
x-y=3
2x-2y=k
The linear system 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.
True
If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
False
If each equation is 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.
False
Elementary row ops permit one equation in a linear system to be subtracted from another.
True
If a matrix is in reduced row echelon form, then it is also in row echelon form.
True
If an elementary row op is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.
False
Every matrix has a unique row echelon form.
False
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.
True
All leading 1's in a matrix in row echelon form must occur in different columns.
True
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.
False
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.
True
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.
False
If a linear system has more unknowns than equations, then it must have infinitely many solutions.
False
If A and B are 2x2 matrices, then AB=BA.
False
The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B.
False
tr(AB) = tr(A)tr(B)
False
(AB)transpose=(A)trans(b)trans
False
tr(A trans)=tr(A)
True
tr(cA)=ctr(A)
True
AC=BC, then A=B
False
If B has a column of zeros, then so does AB if this product is defined.
True
If B has a column of vectors, then so down BA if this product is defined.
False
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.
True
(AB)trans=(B)trans(A)trans
True
The transpose of a product of any number of matrices is the product of the transposes in the reverse order
True
two nxn matrices are inverses of one another iff AB=BA=0
False
A^2-B^2=(A-B)(A+B)
False
(AB)^-1=A^-1B^-1
False
(kA+B)trans=k(A)trans + (B)trans
True
If A is an invertible matrix, then so is A trans
True
A square matrix containing a row or column of zeros cannot be invertible.
True
The sum of two invertible matrices of the same size must be invertible.
False
The product of two elementary matrices of the same size must be an elementary matrix.
False
Every elementary matrix is invertible.
True
If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent.
True
If A is an nxn matrix that is not invertible, then the linear system Ax=0 has infinitely many solutions.
True
If A is an nxn matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible.
True
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.
True
An expression of the invertible matrix A as a product of elementary matrices is unique.
False
The inverse of an inbertible lower triangular matrix is an upper triangular matrix
False
A diagonal matrix is invertible iff all of its diagonal entries are positive.
False.
A matrix that is both symmetric and upper triangular must be a diagonal matrix.
True
A + B is symmetric, then A and B are symmetric.
False
A + B is upper triangular, then A and B are upper triangular.
False
det(A+B)=det(A) + det(B)
False
det(AB) = det(A)det(B)
True
det(A^-1)=1/det(A)
True
Every linearly independent subset of a vector space V is a basis for V.
False
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
True.
The cooridnate vector of a vector x in Rn relative to the standard basis for Rn is x.
True
Every basis of P4 contains at least one polynomial of degree 3 or less.
False
A vector space that cannot be spanned by finitely many vectors is said to be infinite-dimensional.
True
A set containing a single vector is linearly independent.
False
The set of vectors (v,kv) is linearly dependent for every scalar k.
True
Every linearly dependent set contains the zero vector.
False
If the set of vectors is linearly independent, then kv1, kv2, kv2 is also linearly independent for every nonzero scalar k.
True
If v1, ...., vn are linearly dependent nonzero vectors, then at least one vector vk is a unique linear combination of v1, ..., v(k-1)
True
The set of 2x2 matrices that contain exactly two 1's and two 0's is a linearly independent in M22.
False
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.
False
A set with exactly one vector is linearly independent if and only if that vector is not 0.
True
A set with exactly two vectors is linearly independent if and only if neither vector is a scalar multiple of the other.
True
A finite set that contains 0 is linearly dependent.
True
A vector is a directed line segment (an arrow).
False
A vector is an n-tuple of real numbers.
False
A vector is any element of a vector space.
True
There is a vector space consisting of exactly two distinct vectors.
False
The set of polynomials with degree exactly 1 is a vector space under the operations.
False
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