True or False Flashcards
A vector is any element of a vector space.
TRUE
A vector space must contain at least two vectors.
FALSE
If u is a vector and k is a scalar such that ku = 0, then it must be true that k = 0.
FALSE
The set of positive real numbers is a vector space if vector addition and scalar multiplication are the usual operations of addition and multiplication of real numbers.
FALSE
In every vector space the vectors (โ1)u and โu are the
same.
TRUE
In the vector space ๐น(โโ, โ) any function whose graph
passes through the origin is a zero vector.
FALSE
Every subspace of a vector space is itself a vector space.
TRUE
Every vector space is a subspace of itself.
TRUE
Every subset of a vector space ๐ that contains the zero vector in ๐ is a subspace of ๐.
FALSE
The kernel of a matrix transformation ๐๐ด โถ ๐
n โ๐
m is a
subspace of ๐
m.
FALSE
The solution set of a consistent linear system ๐ดx = b of m equations in n unknowns is a subspace of ๐ n.
FALSE
The intersection of any two subspaces of a vector space ๐
is a subspace of ๐.
TRUE
The union of any two subspaces of a vector space ๐ is a subspace of ๐.
FALSE
The set of upper triangular n ร n matrices is a subspace of
the vector space of all n ร n matrices.
TRUE
An expression of the form k1v1 + k2v2 + โ โ โ krvr is called a linear combination.
TRUE
The span of a single vector in ๐ 2 is a line.
FALSE
The span of two vectors in ๐ 3 is a plane.
FALSE
The span of a nonempty set ๐ of vectors in ๐ is the smallest subspace of ๐ that contains ๐.
TRUE
The span of any finite set of vectors in a vector space is closed under addition and scalar multiplication.
TRUE
Two subsets of a vector space ๐ that span the same subspace of ๐ must be equal.
FALSE
The polynomials x โ 1, (x โ 1)2, and (x โ 1)3 span ๐3.
FALSE
A set containing a single vector is linearly independent.
FALSE
No linearly independent set contains the zero vector.
TRUE
Every linearly dependent set contains the zero vector.
FALSE
If the set of vectors {v1, v2, v3} is linearly independent, then {kv1, kv2, kv3} 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, . . . , vkโ1.
TRUE
The set of 2 ร 2 matrices that contain exactly two 1โs and two 0โs is a linearly independent set in ๐22
FALSE
The three polynomials (x โ 1)(x + 2), x(x + 2), and
x(x โ 1) are linearly independent.
TRUE
The functions ๐1 and ๐2 are linearly dependent if there is a real number x such that k1๐1(x) + k2๐2(x) = 0 for some scalars k1 and k2.
FALSE
If ๐ = span{v1, . . . , vn}, then {v1, . . . , vn} is a basis for ๐
FALSE
Every linearly independent subset of a vector space ๐ is a basis for ๐.
FALSE
If {v1, v2, . . . , vn} is a basis for a vector space ๐, then every vector in ๐ can be expressed as a linear combination of v1, v2, . . . , vn
TRUE
The coordinate vector of a vector x in ๐ n relative to the standard basis for ๐ n is x
TRUE
Every basis of ๐4 contains at least one polynomial of
degree 3 or less
FALSE
The zero vector space has dimension zero.
TRUE
There is a set of 17 linearly independent vectors in ๐ 17
TRUE
There is a set of 11 vectors that span ๐ 17
FALSE
Every linearly independent set of five vectors in ๐ 5 is a basis for ๐ 5
TRUE
Every set of five vectors that spans ๐ 5 is a basis for ๐ 5
TRUE
Every set of vectors that spans ๐ n contains a basis for ๐ n
TRUE
Every linearly independent set of vectors in ๐ n is contained in some basis for ๐ n
TRUE
There is a basis for ๐22 consisting of invertible matrices
TRUE
If ๐ด has size n ร n and ๐ผn, ๐ด, ๐ด2, . . . , ๐ดn2 are distinct
matrices, then {๐ผn, ๐ด, ๐ด2, . . . , ๐ดn2 } is a linearly dependent set
TRUE
There are at least two distinct three-dimensional subspaces of ๐2
FALSE
There are only three distinct two-dimensional subspaces of ๐2.
FALSE
If ๐ต1 and ๐ต2 are bases for a vector space ๐, then there exists a transition matrix from ๐ต1 to ๐ต2
TRUE
Transition matrices are invertible
TRUE
If ๐ต is a basis for a vector space ๐ n, then ๐๐ตโ๐ต is the identity matrix
TRUE
If ๐๐ต1โ๐ต2 is a diagonal matrix, then each vector in ๐ต2 is a scalar multiple of some vector in ๐ต1
TRUE
If each vector in ๐ต2 is a scalar multiple of some vector in ๐ต1, then ๐๐ต1โ๐ต2 is a diagonal matrix
FALSE
If ๐ด is a square matrix, then ๐ด = ๐๐ต1โ๐ต2 for some bases ๐ต1 and ๐ต2 for ๐ n
FALSE
The span of v1, . . . , vn is the column space of the matrix whose column vectors are v1, . . . , vn.
TRUE
The column space of a matrix ๐ด is the set of solutions of ๐ดx = b
FALSE
If ๐ is the reduced row echelon form of ๐ด, then those column vectors of ๐ that contain the leading 1โs form a basis for the column space of ๐ด.
FALSE
The set of nonzero row vectors of a matrix ๐ด is a basis for the row space of ๐ด.
FALSE
If ๐ด and ๐ต are n ร n matrices that have the same row
space, then ๐ด and ๐ต have the same column space
FALSE
If ๐ธ is an m ร m elementary matrix and ๐ด is an m ร n
matrix, then the null space of ๐ธ๐ด is the same as the null space of ๐ด
TRUE
If ๐ธ is an m ร m elementary matrix and ๐ด is an m ร n
matrix, then the row space of ๐ธ๐ด is the same as the row space of ๐ด
TRUE
If ๐ธ is an m ร m elementary matrix and ๐ด is an m ร n
matrix, then the column space of ๐ธ๐ด is the same as the column space of ๐ด
FALSE
The system ๐ดx = b is inconsistent if and only if b is not
in the column space of ๐ด.
TRUE
There is an invertible matrix ๐ด and a singular matrix ๐ต
such that the row spaces of ๐ด and ๐ต are the same
FALSE
Either the row vectors or the column vectors of a square matrix are linearly independent
FALSE
A matrix with linearly independent row vectors and linearly independent column vectors is square
TRUE
The nullity of a nonzero m ร n matrix is at most m
FALSE
Adding one additional column to a matrix increases its rank by one
FALSE
The nullity of a square matrix with linearly dependent
rows is at least one
TRUE
If ๐ด is square and ๐ดx = b is inconsistent for some vector b, then the nullity of ๐ด is zero
FALSE
If a matrix ๐ด has more rows than columns, then the
dimension of the row space is greater than the dimension of the column space
FALSE
If rank(๐ด๐) = rank(๐ด), then ๐ด is square
FALSE
There is no 3 ร 3 matrix whose row space and null space are both lines in 3-space
TRUE
If ๐ is a subspace of ๐ n and ๐ is a subspace of ๐, then ๐โ is a subspace of ๐โ
FALSE
If ๐ด is a square matrix and ๐ด x = ๐x for some nonzero
scalar ๐, then x is an eigenvector of ๐ด.
FALSE
If ๐ is an eigenvalue of a matrix ๐ด, then the linear system (๐๐ผ โ ๐ด)x = 0 has only the trivial solution
FALSE
If the characteristic polynomial of a matrix ๐ด is
p(๐) = ๐2 + 1
then ๐ด is invertible
TRUE
If ๐ is an eigenvalue of a matrix ๐ด, then the eigenspace of ๐ด corresponding to ๐ is the set of eigenvectors of ๐ด corresponding to ๐
FALSE
The eigenvalues of a matrix ๐ด are the same as the eigenvalues of the reduced row echelon form of ๐ด
FALSE
If 0 is an eigenvalue of a matrix ๐ด, then the set of columns of ๐ด is linearly independent
FALSE
An n ร n matrix with fewer than n distinct eigenvalues is not diagonalizable
FALSE
An n ร n matrix with fewer than n linearly independent eigenvectors is not diagonalizable
TRUE
If ๐ด and ๐ต are similar n ร n matrices, then there exists an invertible n ร n matrix ๐ such that ๐๐ด = ๐ต๐
TRUE
If ๐ด is diagonalizable, then there is a unique matrix ๐ such that ๐โ1๐ด๐ is diagonal
FALSE
If ๐ด is diagonalizable and invertible, then ๐ดโ1 is diagonalizable
TRUE
If ๐ด is diagonalizable, then ๐ด๐ is diagonalizable
TRUE
If there is a basis for ๐ n consisting of eigenvectors of an n ร n matrix ๐ด, then ๐ด is diagonalizable
TRUE
If every eigenvalue of a matrix ๐ด has algebraic multiplicity 1, then ๐ด is diagonalizable
TRUE
If 0 is an eigenvalue of a matrix ๐ด, then ๐ด2 is singular
TRUE
The matrix [1 0
0 1
0 0] is orthogonal
FALSE
The matrix [1 โ2
2 1] is orthogonal
FALSE
An m ร n matrix ๐ด is orthogonal if ๐ด๐๐ด = ๐ผ
FALSE
A square matrix whose columns form an orthogonal set is orthogonal
FALSE
Every orthogonal matrix is invertible
TRUE
If ๐ด is an orthogonal matrix, then ๐ด2 is orthogonal and (det ๐ด)2 = 1
TRUE
Every eigenvalue of an orthogonal matrix has absolute value 1
TRUE
If ๐ด is a square matrix and โ๐ดuโ = 1 for all unit vectors
u, then ๐ด is orthogonal
TRUE
If ๐ด is a square matrix, then ๐ด๐ด๐ and ๐ด๐๐ด are orthogonally diagonalizable
TRUE
If v1 and v2 are eigenvectors from distinct eigenspaces of a symmetric matrix with real entries, then
โv1 + v2โ2 = โv1โ2 + โv2โ2
TRUE
Every orthogonal matrix is orthogonally diagonalizable
FALSE
If ๐ด is both invertible and orthogonally diagonalizable,
then ๐ดโ1 is orthogonally diagonalizable
TRUE
Every eigenvalue of an orthogonal matrix has absolute value 1.
TRUE
If ๐ด is an n ร n orthogonally diagonalizable matrix, then there exists an orthonormal basis for ๐ n consisting of eigenvectors of ๐ด.
TRUE
If ๐ด is orthogonally diagonalizable, then ๐ด has real eigenvalues.
TRUE
If all eigenvalues of a symmetric matrix ๐ด are positive,
then ๐ด is positive definite
TRUE
x2^1โ x2^2 + x3^2 + 4x1x2x3 is a quadratic form
FALSE
(x1 โ 3x2)^2 is a quadratic form
TRUE
A positive definite matrix is invertible
TRUE
A symmetric matrix is either positive definite, negative definite, or indefinite
FALSE
If ๐ด is positive definite, then โ๐ด is negative definite
TRUE
x ยท x is a quadratic form for all x in ๐ n
TRUE
If ๐ด is symmetric and invertible, and if x๐๐ดx is a positive definite quadratic form, then x๐๐ดโ1x is also a positive definite quadratic form
TRUE
If ๐ด is symmetric and has only positive eigenvalues, then x๐๐ดx is a positive definite quadratic form
TRUE
If ๐ด is a 2 ร 2 symmetric matrix with positive entries and det(๐ด) > 0, then ๐ด is positive definite
TRUE
If ๐ด is symmetric, and if the quadratic form x๐๐ดx has no cross product terms, then ๐ด must be a diagonal matrix
TRUE
If x๐๐ดx is a positive definite quadratic form in two variables and c โ 0, then the graph of the equation x๐๐ดx = c is an ellipse
FALSE