true. If a multiple of one row of A is added to another row to produce a matrix B, then detB=detA

a row replacement operation does not affect the determinant of a matrix

true

the determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U.

true. also when the rows of A are linearly dependent. a square matrix is invertible iff detA≠0

if the columns of A are linearly dependent, then detA=0

false. warning det(A+B)≠ detA+detB

det(A+B)=detA + detB

true. 2 interchanges detB= -detA, 3 interchanges detB= detA

if 3 row interchanges are made in succession, then the new determinant equals the old determinant

false. only in row echelon form

the determinant of A is the product of the diagonal entries in A

false

if detA is 0, the two rows or two columns are the same, or a row or a column is 0

false detA⁻¹= 1/detA

detA⁻¹=(-1)detA

false. for ALL t

if f is a function in the vector space V of all real-valued function on R and if f(t)=0 for some t, then f is the zero vector in V

false, can be any dimension

a vector is an arrow in 3-D space

false, needs to be closed under addition and multiplication

a subset H of a vector space V is a subspace of V if the zero vector in in H

true

a subspace is also a vector space

true

a vector is an element of a vector space

true. -u=(-1)u

if u is a vector in a vector space V, the (-1)u is the same as the negative of u

false. R² is not a subspace of R³ cuz R² is not even a subset of R³

R² is a subspace of R³

false. part (ii) does

not state that u and v represent all possible elements of H

not state that u and v represent all possible elements of H

a subset of H of a vector space V is a subspace of V if the following conditions are met:

i. the zero vector of V is in H

ii. u,v, and u+v are in H

iii. c is a scalar and cu is in H

i. the zero vector of V is in H

ii. u,v, and u+v are in H

iii. c is a scalar and cu is in H

true. the null space is the set of all solutions of the homogeneous equation Ax=0

the null space of A is the solution set of the equation Ax=0

false null space= Rⁿ

the null space of an mxn matriz if R^m

true

the column space of A is the range of the mapping x-> Ax

false. The equation Ax = b must be consistent for every b

if the equation Ax=b is consistent, then ColA=R^m

true

the kernel of a linear transformation is a vector space

true. a vector in ColA can be written as Ax for some x because the notation Ax stands for the linear combination of the columns of A

ColA is the set of all vectors that can be written as Ax for some x

true

a null space is a vector space

true

the column space if an nxm matrix is in R^m

false. for some x in Rⁿ

ColA is the set of all solutions of Ax=b

true

NulA is the kernel of the mapping x->Ax

true

the range of a linear transformation is a vector space

true. NulA= set of all solutions to Ax=0, NulA=kernel A

the set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation

false. only the zero vector is linearly dependent

a single vector by itself is linearly dependent

false. {b1…bp} must also be linearly independent

if H=Span{b1,…bp} then {b1…bp} is a basis for H

true. invertible means columns are linearly independent and they span Rⁿ

the columns of an invertible nxn matrix for a basis for Rⁿ

false. as small as possible

a basis is a spanning set that is as large as possible

false. EROs don’t change the linear dependence relations

in some cases the linear dependence relations amonng the columns of a matrix can be affected by certain elementary row operations on the matrix

false. The subspace spanned by the set must also coincide with H

a linearly independent set in a subspace H is a basis for H

true. Apply the Spanning Set Theorem to V instead of H. The space V is nonzero because the

spanning set uses nonzero vectors.

spanning set uses nonzero vectors.

if a finite set of S of nonzero vectors spans a vector space V, then some subset of S is a basis for V

true

a basis is a linearly independent set that is as large as possible

false. method always works

the standard method for producing a spanning set for NulA sometimes fails to produce a basis for NulA

false, pivot columns of A form a basis for ColA

if B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA

true. if c1…cn are the B-coordinates of x, then the vector [x]B= [c1 … cn]

if x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rⁿ

false

if PB is the change of coordinates matrix, then [x]B=PBx for x in V

false P3 is isomorphic to R4

the vector spaces P3 and R³ are isomorphic

true

if B is the standard basis for Rⁿ then the B-coordinate vector of an x in Rⁿ is x itself

false. coordinate mapping goes in the opposite direction x->[x]b

the correspondence [x]b->x is called the coordinate mapping

true. If the plane passes through the origin the plane is isomorphic to R2

in some cases, a plane in R³ can be isomorphic to R²

true

the number of pivot columns of a matrix equals the dimension of the column space

false. The plane must pass through the origin

a plane in R3 is a 2-D subspace of R3

false dim=5

the dimension of the vector space P4 is 4

false. The set S must also have n elements. any linearly independent set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V

if dimV=n and S is a linearly independent set in V, then S is a basis for V

true. if a vector space V has a basis B then any set in V containing more than n vectors must be lin dep

if a set {v1…vp} spans a finite dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.

false, R2 is not a subset of R3

R2 is a 2-D subspace of R3

false. number of free variables=dimNulA

the number of variables in the equation Ax=0 equals the dimension of NulA

false. A basis could still have only finitely many elements, which would make the vector space finitedimensional

a vector space is infinite dimensional if it is spanned by an infinite set

false. S needs n elements

if dimV=n and if S spans V, then S is a basis of V

true

the only 3-D subspace of R3 is R3 itself

true. Apply the Spanning Set Theorem to the set {v1, , vp} and produce a basis for V. This basis will not have more than p elements in it, so dimV ≤ p.

if there exists a set {v1…vp} that spans V, then dimV<=p

true. any linearly ind set in H can be expanded to a basis for H. This basis will have at least

p elements in it, so dimV ≥ p.

dimH≤dimV

p elements in it, so dimV ≥ p.

dimH≤dimV

if there exists a lin ind set {v1..vp} in V, the dimV>=p

true. Take any basis (which will contain p vectors) for V and adjoin the zero vector to it.

if DimV=p then there exists a spanning set of p+1 vectors in V

false

if there exists a lin dep set in V, then dimV≤p

true. If dimV ≤ p, there is a basis for V with p or fewer vectors. This basis would be a spanning set

for V with p or fewer vectors,

for V with p or fewer vectors,

if every set of p elements in V fails to span V then dimV>p

false

if p≥2 and dimV=p then every set of p-1 nonzero vectors is lin ind