Linear Algebra Part 1

Front 1

A linear transformation T : Rn ! Rm is completely
determined by its eect on columns of the n n identity
matrix

Back 1

TRUE The columns on the identity matrix are the
basis vectors in Rn. Since every vector can be written as a
linear combination of these, and T is a linear transformation,
if we know where these columns go, we know everything.

Front 2

I If T : R2 ! R2
rotates vectors about the origin through an
angle , then T is a linear transformation

Back 2

n. TRUE. To show
this we would show the properties of linear transformations
are preserved under rotations.

Front 3

When two linear transformations are performed one after
another, then combined eect may not always be a linear
transformation

Back 3

FALSE Again, check properties to show it is a
linear transformation.

Front 4

The columns of the standard matrix for a linear
transformation from R
n
to R
m are the images of the columns
of the n n identity matrix.

Back 4

TRUE

Front 5

The standard matrix of a linear transformation from R
2
to R
2
that reflects points through the horizontal axis, the vertical axis, or the origin has the form 
a 0
0 d

where a and d are +/- 1

Back 5

TRUE We can check this by checking the images of the
basis vectors.

Front 6

A mapping T : Rn ! Rm is one-to-one if each vector in R
n maps onto a unique vector in Rm

Back 6

FALSE A mapping is
one-to-one if each vectors in R
m is mapped to from a unique
vector in Rn

Front 7

If A is a 3 2 matrix, then the transformation x ! Ax cannot
map R
2 onto R
3

Back 7

TRUE You can not map a space of lower
dimension ONTO a space of higher dimension.

Front 8

A homogeneous equation is always consistent.

Back 8

TRUE - The
trivial solution is always a solution.

Front 9

The equation Ax = 0 gives an explicit descriptions of its
solution set.

Back 9

FALSE - The equation gives an implicit
description of the solution set.

Front 10

The homogeneous equation Ax = 0 has the trivial solution if
and only if the equation has at least one free variable

Back 10

FALSE
- The trivial solution is always a solution to the equation
Ax = 0.

Front 11

The equation x = p + tv describes a line through v parallel to
p

Back 11

False. The line goes through p and is parallel to v.

Front 12

The solution set of Ax = b is the set of all vectors of the form
w = p + vh where vh is any solution of the equation Ax = 0

Back 12

FALSE This is only true when there exists some vector p such
that Ap = b.

Front 13

If x is a nontrivial solution of Ax = 0, then every entry in x is
nonzero

Back 13

FALSE. At least one entry in x is nonzero

Front 14

The equation x = x2u + x3v, with x2 and x3 free (and neither
u or v a multiple of the other), describes a plane through the
origin

Back 14

True

Front 15

The equation Ax = b is homogeneous if the zero vector is a
solution

Back 15

TRUE. If the zero vector is a solution then
b = Ax = A0 = 0. So the equation is Ax = 0, thus
homogenous.

Front 16

The effect of adding p to a vector is to move the vector in the
direction parallel to p.

Back 16

TRUE. We can also think of adding p
as sliding the vector along p.

Front 17

The solution set of Ax = b is obtained by translating the
solution set of Ax = 0

Back 17

FALSE. This only applies to a
consistent system.
Linear Algebra, David

Front 18

The columns of the matrix A are linearly independent if the
equation Ax = 0 has the trivial solution.

Back 18

FALSE. The trivial
solution is always a solution

Front 19

If S is a linearly dependent set, then each vector is a linear
combination of the other vectors in S.

Back 19

FALSE- For example,
[1; 1] , [2; 2] and [5; 4] are linearly dependent but the last is
not a linear combination of the rst two.

Front 20

a four by five matrix has linearly independent columns

Back 20

TRUE. There are five columns each with four entries, thus by
Thm 8 they are linearly dependent

Front 21

If x and y are linearly independent, and if fx; y; zg is linearly
dependent, then z is in Spanfx; yg

Back 21

TRUE Since x and y are
linearly independent, and fx; y; zg is linearly dependent, it
must be that z can be written as a linear combination of the
other two, thus in in their span.

Front 22

Two vectors are linearly dependent if and only if they lie on a
line through the origin

Back 22

TRUE. If they lie on a line through
the origin then the origin, the zero vector, is in their span thus
they are linearly dependent

Front 23

If a set contains fewer vectors then there are entries in the
vectors, then the set is linearly independent.

Back 23

FALSE For
example, [1; 2; 3] and [2; 4; 6] are linearly dependent.

Front 24

FALSE For
example, [1; 2; 3] and [2; 4; 6] are linearly dependent.

Back 24

TRUE If z is
in the Spanfx; yg then z is a linear combination of the other
two, which can be rearranged to show linear dependence.

Front 25

I If a set in R
n
is linearly dependent, then the set contains more
vectors than there are entries in each vector.

Back 25

False. For
example, in R
3
[1; 2; 3] and [3; 6; 9] are linearly dependent.

Front 26

A linear transformation is a special type of function

Back 26

TRUE
The properties are (i) T(u + v) = T(u) + T(v) and (ii)
T(cu) = cT(u).

Front 27

If A is a 3 5 matrix and T is a transformation dened by
T(x) = Ax, then the domain of T is R
3

Back 27

FALSE The domain
is R
5

Front 28

If A is an m n matrix, then the range of the transformation
x 7! Ax is R
m

Back 28

FALSE R
m is the codomain, the range is where
we actually land.

Front 29

A transformation T is linear if and only if
T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the
domain of T and for all scalars c1 and c2.

Back 29

TRUE If we take
the denition of linear transformation we can derive these and
if these are true then they are true for c1; c2 = 1 so the rst
part of the denition is true, and if v = 0, then the second
part if true.

Front 30

Every matrix transformation is a linear transformation.

Back 30

TRUE
To actually show this, we would have to show all matrix
transformations satisfy the two criterion of linear
transformations.

Front 31

The codomain of the transformation x 7! Ax is the set of all linear combinations of the columns of A

Back 31

FALSE The If A is
m n codomain is R
m. The original statement in describing
the range

Front 32

If T : Rn ! R
m is a linear transformation and if c is in Rm,
then a uniqueness question is "Is c is the range of T."

Back 32

FALSE
This is an existence question.

Front 33

A linear transformation preserves the operations of vector
addition and scalar multiplication

Back 33

TRUE This is part of the
denition of a linear transformation

Front 34

The superposition principle is a physical description of a linear
transformation.

Back 34

TRUE The book says so. (page 77)

Front 35

In some cases a matrix may be row reduced to more than one
matrix in reduced row echelon form, using different sequences
of row operations.

Back 35

FALSE

Front 36

The row reduction algorithm applies only to augmented
matrices for a linear system.

Back 36

FALSE

Front 37

A basic variable in a linear system is a variable that
corresponds to a pivot column in the coecient matrix.

Back 37

TRUE

Front 38

Finding a parametric description of the solution set of a linear
system is the same as solving the system.

Back 38

TRUE

Front 39

If one row in an echelon form of an augmented matrix is
[0 0 0 5 0 ], then the associated linear system is inconsistent.

Back 39

FALSE

Front 40

The echelon form of a matrix is unique

Back 40

FALSE

Front 41

The pivot positions in a matrix depend on whether row
interchanges are used n the row reduction process.

Back 41

FALSE

Front 42

Reducing a matrix to echelon form is called the forward phase
of the row reduction process

Back 42

TRUE

Front 43

Whenever a system has free variables, the solution set
contains many solutions

Back 43

FALSE

Front 44

A general solution os a system is an explicit description os all
solutions of the system

Back 44

TRUE

Front 45

An example of a linear combination of vectors v1 and v2 is the
vector 1=2v1

Back 45

TRUE

Front 46

The solution set of the linear system whose augmented matrix
is [a1 a2 a3 b] is the same as the solution set of the equation
x1a1 + x2a2 + x3a3 = b

Back 46

TRUE

Front 47

The set Span U,V is always visualized as a plane through
the origin.

Back 47

FALSE

Front 48

Any list of FIve real numbers is a vector in R
5

Back 48

TRUE

Front 49

The weights c1; : : : ; cp is a linear combination
c1v1 +


+ cpvp cannot all be zero

Back 49

FALSE

Front 50

When u and v are nonzero vectors, Span U, V contains the
line through u and the origin.

Back 50

TRUE

Front 51

Asking whether the linear system corresponding to an
augmented matrix [a1 a2 a3 b] has a solution amounts to
asking whether b is in Span a1; a2; a3

Back 51

TRUE

Front 52

The equation Ax = b is referred to as the vector equation.

Back 52

FALSE

Front 53

The vector b is a linear combination of the columns of a
matrix A if and only if the equation Ax = b has at least one
solution

Back 53

TRUE

Front 54

The equation Ax = b is consistent if the augmented matrix
[A b] has a pivot position in every row

Back 54

FALSE

Front 55

The rst entry in the product Ax is a sum of products.

Back 55

TRUE

Front 56

If the columns of an m n matrix span R
m, then the equation
Ax = b is consistent for each b in R
m

Back 56

TRUE

Front 57

I If A is an m n matrix and if the equation Ax = b is
inconsistent for some b in R
m, then A cannot have a pivot
position in every row.

Back 57

TRUE

Front 58

Every matrix equation Ax = b corresponds to a vector
equation with the same solution set

Back 58

TRUE

Front 59

Any linear combination of vectors can always be written in the
form Ax for a suitable matrix A and vector x.

Back 59

TRUE

Front 60

The solution set of the linear system whose augmented matrix
is [a1 a2 a3 b] is the same as the solution set of Ax = b, if
A = [a1 a2 a3

Back 60

TRUE

Front 61

If the equation Ax = b is inconsistent, then b is not in the set
spanned by the columns of A.

Back 61

TRUE

Front 62

If the augmented matrix [A b] has a pivot position in every
row, then the equation Ax = b is inconsistent.

Back 62

FALSE

Front 63

If A is an m n matrix whose columns do not span R
m, then
the equation Ax = b is inconsistent for some b in R
m

Back 63

TRUE

Front 64

I If A and B are 2 2 with columns a1; a2 and b1; b2 then
AB = [a1b1; a2b2]

Back 64

FALSE Matrix multiplication is "row by
column".

Front 65

Each column of AB is a linear combination of the columns of
A using weights from the corresponding column of B

Back 65

FALSE
Swap A and B then its true

Front 66

AB + AC = A(B + C)

Back 66

TRUE Matrix multiplication distributes
over addition.

Front 67

A^T + B^T = (A + B)^T

Back 67

TRUE See properties of transposition.
Also should be able to think through to show this. When we
add we add corresponding entries, these will remain
corresponding entries after transposition.

Front 68

The transpose of a product of matrices equals the product of
their tranposes in the same order

Back 68

FALSE The transpose of a
product of matrices equals the product of their tranposes in
the reverse order.
Linear Algebra, David Lay Week F

Front 69

If A and B are 3 3 and B = [b1 b2 b3], then
AB = [Ab1 + Ab2 + Ab3].

Back 69

FALSE This is right but there
should not be +'s in the solution. Remember the answer
should also be 3 3.

Front 70

The second row of AB is the second row of A multiplied on
the right by B

Back 70

TRUE

Front 71

(AB)C = (AC)B

Back 71

FALSE Matrix multiplication is not
commutative

Front 72

The transpose of a sum of matrices equals the sum of their
transposes

Back 72

TRUE

Front 73

In order for a matrix B to be the inverse of A, both equations
AB = I and BA = I must be true.

Back 73

TRUE We'll see later that
for square matrices AB=I then there is some C such that
BC=I. CHALLENGE: Can you FInd an inverse for any
non-square matrix. If so FInd one, if not explain why. Also in
above statement about square matrices, does C=A?

Front 74

If A is an invertible n n matrix, then the equation Ax = b is
consistent for each b in R
n
.

Back 74

TRUE. Since A is invertible we
have that X=(A^-1)b

Front 75

Each elementary matrix is invertible

Back 75

True. Let K be the
elementary row operation required to change the elementary
matrix back into the identity. If we preform K on the identity,
we get the inverse.

Front 76

A product of invertible n n matrices is invertible, and the
inverse of the product of their matrices in the same order.

Back 76

FALSE. It is invertible, but the inverses in the product of the
inverses in the reverse order

Front 77

FALSE. It is invertible, but the inverses in the product of the
inverses in the reverse order

Back 77

TRUE

Front 78

I If A =

a b
c d 
and ad = bc, then A is not invertible

Back 78

I If A =

a b
c d 
and ad = bc, then A is not invertible

Front 79

If A can be row reduced to the identity matrix, then A must
be invertible.

Back 79

TRUE The algorithm presented in this chapter
tells us how to nd the inverse in this case.

Front 80

If A is invertible, then elementary row operations then reduce
A to to the identity also reduce A^-1 to the identity

Back 80

FALSE
They also reduce the identity to A^-1

Front 81

If the equation Ax = 0 has only the trivial solution, then A is
row equivalent to the n n identity matrix.

Back 81

TRUE From Thm
8

Front 82

If the columns of A span R
n
, then the columns are linearly
independent

Back 82

TRUE Again from Thm 8. Also is n vector span
R
n
they must be linearly independent.

Front 83

If A is an n n matrix then the equation Ax = b has at least
one solution for each b in R
n
.

Back 83

FALSE we need to know more
about A like if it in invertible (or anything else in Thm 8)

Front 84

If the equation Ax = 0 has a nontrivial solution, then A has
fewer than n pivot positions.

Back 84

TRUE This comes from the "all
false" part of THM 8. (The statements are either all true or
all false.)

Front 85

If A^T is not invertible, then A is not invertible

Back 85

TRUE Also
from the all false part of theorem 8

Front 86

sd;lkjWEI

Back 86

TRUE Thm 8

Front 87

If the columns of A are linearly independent, then the columns
of A span R
n
.

Back 87

TRUE Thm 8

Front 88

If the equation Ax = b has at least solution for each b in R
n
,
then the solution is unique for each b.

Back 88

TRUE Thm 8

Front 89

If the linear transformation x 7! Ax maps R
n
into R
n
then A
has n pivot points.

Back 89

FALSE. Since A is n n the linear
transformation x 7! Ax maps R
n
into R
n
. This doesn't tell us
anything about A.

Front 90

If there is a b in R
n
such that the equation Ax = b is
inconsistent, then the transformation x 7! Ax is not
one-to-one.

Back 90

TRUE Thm 8

Front 91

The (i; j)-cofactor of a matrix A is the matrix Aij obtained by
deleting from A its ith row and jth column.

Back 91

FALSE The
cofactor is the determinant of this Aij times -1^(i+j)

Front 92

The cofactor expansion of det A down a column is the
negative of the cofactor expansion along a row

Back 92

FALSE We
can expand down any row or column and get same
determinant

Front 93

The determinant of a triangular matrix is the sum of the
entries of the main diagonal.

Back 93

FALSE It is the product of the
diagonal entries

Front 94

A row replacement operation does not aect the determinant
of a matrix.

Back 94

TRUE Just make sure you don't multiply the row
you are replacing by a constant.

Front 95

If the columns of A are linearly dependent, then det A = 0.

Back 95

TRUE (For example there is a row without a pivot so must be
a row of all zeros.)

Front 96

det(A + B)= detA + detB.

Back 96

FALSE This is true for product
however.

Front 97

If two row interchanges are made in succession, then the new
determinant equals the old determinant

Back 97

TRUE Both changes
multiply the determinant by -1 and -1*-1=1.

Front 98

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

Back 98

FALSE unless A is triangular.

Front 99

If det A is zero, then two rows or two columns are the same,
or a row or a column is zero.

Back 99

FALSE The converse is true,
however.

Front 100

det(A^t)= -1detA

Back 100

FALSE det(A^T)=det A when A is nxn

Front 101

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

Back 101

FALSE We need f(t) = 0 for all t.

Front 102

A vector is an arrow in three-dimensional space

Back 102

FALSE This
is an example of a vector, but there are certainly vectors not
of this form.

Front 103

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

Back 103

FALSE We also need the set to be closed under
addition and scalar multiplication.

Front 104

A subspace is also a vector space

Back 104

TRUE This is the definition
of subspace, a subset that satisfies the vector space properties.

Front 105

Analogue signals are used in the major control systems for the
space shuttle, mentioned in the introduction to the chapter.

Back 105

FALSE Digital signals are used...