LINEAR ALGEBRA EXAM 2 T/F QUESTIONS AND ANSWERS

LINEAR ALGEBRA EXAM 2 T/F
QUESTIONS AND ANSWERS
If f is a function in the vector space V of all real-valued functions on R and if f(t) = 0 for
some t, then f is the zero vector in V (4.1) - Answer-False - all t

A vector is an arrow in three-dimensional space (4.1) - Answer-False - not every arrow
is a vector

A subset H of a vector space V is a subspace of V if the zero vector is in H (4.1) -
Answer-False - not all subsets that have the zero vector are subspaces

A subspace is also a vector space (4.1) - Answer-True

A vector is any element of a vector space (4.1) - Answer-True

If u is a vector in a vector space V, then (-1)u is the same as the negative of u (4.1) -
Answer-True

A vector space is also a subspace (4.1) - Answer-True

R^2 is a subspace of R^3 (4.1) - Answer-False

A subset H of a vector space V is a subspace of V if the following conditions are
satisfied: (i) the zero vector of V is in H, (ii) u,v, and u+v are in H, and (iii) c is a scalar
and cu is in H. (4.1) - Answer-True

The null space of A is the solution set of the equation Ax = 0 (4.2) - Answer-True

The null space of an mxn matrix is in R^m (4.2) - Answer-False - R^n

The column space of A is the range of the mapping x --> Ax (4.2) - Answer-True

If the equation Ax = b is consistent, then Col A is R^m (4.2) - Answer-False, consistent
for every vector b

The kernel of a linear transformation is a vector space (4.2) - Answer-True

ColA is the set of all vectors that can be written as Ax for some x (4.2) - Answer-True

A null space is a vector space (4.2) - Answer-True

The column space of an mxn matrix is in R^m (4.2) - Answer-True

, ColA is the set of all solutions of Ax=b (4.2) - Answer-False

NulA is the kernel of the mapping x-->Ax (4.2) - Answer-True

The range of a linear transformation is a vector space (4.2) - Answer-True

The set of all solutions of a homogeneous linear differential equation is the kernel of a
linear transformation (4.2) - Answer-True

A single vector by itself is linearly dependent (4.3) - Answer-False - must be zero vector
to be linearly dependent by itself

If H = Span{b1,...,bp}, then {b1,...bp} is a basis for H (4.3) - Answer-False - the set also
has to be linearly independent

The columns of an invertible nxn matrix form a basis for R^n (4.3) - Answer-True

A basis is a spanning set that is as large as possible (4.3) - Answer-False - if it is too
large, it becomes linearly dependent

In some cases, the linear dependence relations among the columns of a matrix can be
affected by certain elementary row operations on the matrix (4.3) - Answer-False

A linearly independent set in a subspace H is a basis for H (4.3) - Answer-False - the
set also has to span H

If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a
basis for V (4.3) - Answer-True

A basis is a linearly independent set that is as large as possible (4.3) - Answer-True

The standard method for producing a spanning set for NulA sometimes fails to produce
a basis for NulA (4.3) - Answer-False

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA
(4.3) - Answer-False - pivot columns of matrix A form a basis for ColA

If x is in V and if B contains n vectors, then the B-coordinate vector of x is in R^n (4.4) -
Answer-True

If P_B is the change-of-coordinates matrix, then [x]_B = P_Bx for x in V (4.4) - Answer-
False - P_B^(-1)x

The vector spaces P_3 and R^3 are isomorphic (4.4) - Answer-False - P_3 is
isomorphic to R^4