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Reference page
Definition.
A vector space
is a set V of elements called vectors that can be
added and scaled so that the following axioms hold true
A1
(commutativity)
A+B=B+A
for each A,B in V
A2
(associativity)
(A+B) + C = A + (B+C)
for each A, B, C in V
A3 (zero
vector)
There is a unique vector 0 such that A+0=A
for each A
in V
A4
(inverse vector)
For each vector A, there is a unique
vector -A
such that A+(-A)=0
A5
(distributivity)
r(A+B)=rA+rB
for each number r
and A,B in V
A6
(distributivity)
(r+s)A=rA+sA
for each numbers r,s and A in V
A7
(compatibility)
(rs)A=r(sA)
for each numbers r,s and A in V
A8 (identity)
1A=A for
each A in V
Definition.
A subset W
of a vector space V
is said to be a subspace of V
if W is a
vector space with the operation defined on V.
True or not?
1. The
empty set is a subspace of every vector space.
2. In any vector space, rA=sA implies that r=s.
Systems of Linear equations. Row reduction.
Elementary moves in
row reduction method
1. interchange any two equations
2.
multiply any equation by a nonzero scalor
3.
add to one equation a multiple of another one
Reduced system
1. the
first nonzero coefficient in each equation, called pivot, is 1
2. each
pivot is the only nonzero coefficient in its column
3. pivots
form an echelon
, i.e., if aij and amn
are pivots and i>m,
then j>n.
Linear combination.
Definition.
A linear
combination of vectors u1, u2, ..., un in a vector space V is any vector v
in V of the
form
a1u1 + a2u2 + ...+ anun
for some numbers
a1, ..., an.
True or not?
1. 0 is
a linear combination of any nonempty set of vectors.
2. In solving systems of linear equations (by the row reduction method)
we can scale any equation.
3. Every system of linear equations has a solution.
4. Let S be
a subset of a vector space V.
Suppose vectors in S
can be added and scaled in S. Then S is
a subspace of V.
5. The empty subset of a vector space is linearly independent.
6. If a subset S
of a vector space contains 0, then S is linearly
dependent.
7. Let
V be a
vector space. Let T
be a subset of V
and S is a
subset of T.
If S is
linearly
dependent, then T
is also linearly dependent.
8.
Let u1, u2, ..., un be vectors in a vector
space V. Suppose
a1u1 + a2u2 + ...+ anun = 0
for some numbers a1, ..., an, not
all zero. Then V
is linearly dependent.
How to prepare for Quiz
1) Look through
homework problems.
Make sure that you can do all
problems. If homework problems seem difficult, do more problems from
the textbook.
2) Read Lecture
Notes.
Make
sure that you understand all proofs and you can reproduce any argument
(at least one question will be of the form "Prove Theorem/Proposition ?
asserting that ... ").
3) Read the
textbook.
You
will not be asked questions that are not covered in lectures. So
reading the textbook is not a requirement. However, it is extremely
helpful and I would recommend that you read it carefully and make sure
that you understand all the arguments.
Bases
Definition.
Let V be a vector space.
Vectors e1, e2, ..., en form a basis for V if
(i) e1, e2, ..., en span V, and
(ii) e1, e2, ..., en are linearly
independent.
Theorem.
Let V be a vector space.
Vectors e1, e2, ..., en form a basis for V if and only
if
(i) e1, e2, ..., en span V, i.e., each
vector v in V can be expressed as
a linear combination of e1, e2, ..., en,
and
(ii) for
each v such an expression is
unique.
L8 last page
Lecture
10
Definition.
A function T: V -> W is linear if it preserves sums
and scalor products:
(a) T(x+y)
= T(x) + T(y) for x,y in V,
(b) T(cx)
= cT(x) for x in V and c in R.
Kernel N(T)
consists of vectors v
in V with T(v)=0.
Image R(T)
consists of vectors w
in W with T(v)=w for some v in V.
True or not? (here
T: V->W is
a linear function)
1. N(T) is
a subspace of V.
2. R(T) is
a subspace of V.
3. If N(T)
contains only 0,
then T:
V-> W is one-to-one.
L9 last page
Lecture
11
True or not? (here
T: V->W is
a linear function)
1. If
V is
of dimension n
and W is of
dimension m,
then each mxn matrix determines
a linear transformation T:V
-> W.
2. Linear transformation can be added and scaled. The set of linear
transformations from V
to W forms
a vector space. (it is denoted L(V,W)).
3. Dimension of L(V,W) is mn.
4. Suppose that
{v1, v2, ..., vn} is a basis of V, and {w1, w2, ..., wn} is a basis of W. Then the
i-th
row of the matrix representation of T
is the coordinate vector of T(vi).
Lecture
12
True or not?
1. If
I:V->V is
the identity transformation, then in any basis {v1, v2, ..., vn} of V, the matrix [I] of I is nxn and is of the
form
1 0 0 ...
0 0 0
0 1 0 ...
0 0 0
0 0 1 ...
0 0 0
.........
... ........
0 0 0 ... 1 0 0
0 0 0 ... 0 1 0
0 0 0 ... 0 0 1
2. A function U: W->V is
said to be inverse of a linear transformation T:V->W if
UT=IV
3. L(V,W) denotes the
vector space of linear transformations V->W.
dim L(V, W) =
dim L(W, V)
but L(V, W) is not the
same as L(W, V).
4. A linear transformation T
is invertible if and only if it is onto and one-to-one.
5. If T is
invertible, then T -1 is also
invertible. Its inverse is T,
i.e.,
(T -1) -1 = T.
Lecture
13
True or not?
1. A
linear transformation T
is invertible if and only if its matrix representation [T]
is invertible.
2. T is one-to-one if
and only if N(T)={0}.
3. Let T: V->W be a
linear transformation of vector spaces of dimension n.
Then N(T)={0}
if and only if R(T)=W.
4. If AB=I,
then A and B are invertible
matrices.
5. Suppose that A
and B are nxn matrices. Suppose
that AB=I,
then
(a) LB
is one-to-one,
(b) LB is onto. So B is
invertible.
(c) LA is onto,
(d) LA
is one-to-one. So A is invertible.
Lecture
14
True
or not?
1.
A function T is one-to-one if T(v)=T(w) implies v=w
(i.e.,
images of two vectors v and w are the same if and only if v=w).
2. A function T: V-> W is onto if the image of T is all W (i.e.,
each vector w in W is the image of some vector v in V).
3. Let {v1, v2, v3} be a basis of a vector space V.
It is the same as the basis {v2, v1, v3}.
4. Let {v1, v2, v3} be an ordered basis of a vector
space V. It is the same as the ordered basis {v2, v1, v3}.
Lecture
15
True
or not?
1.
Let V and W be vector spaces
of dimensions n
and m with
fixed ordered bases. Then each mxn matrix A corresponds to a
linear transformation T:
V->W. The transformation T is usually denoted
by LA.
2.
Performing a row operation on a matrix A is equivalent to
taking the product of A
and an elementary matrix E;
in both cases the result is a new matrix B=EA.
3. Each elementary matrix is invertible.
Lecture
16
True
or not?
1.
For each matrix A,
there are invertible matrices B
and C such
that BAC is
a matrix D
with Dii = 1 for i<rank(A)
and Dij=0 otherwise.
2. Each matrix A
can be trasformed to a matrix D
as above by elementary row operations.
Lecture
17
True
or not?
1. Let Ax=0
be a system of m
linear equations in n
unknowns. Then the set of solutions
form a subspace of Rn.
2. Let Ax=b be a system of m linear equations
in n
unknowns. Then the set of solutions
form a subspace of Rn.
Lecture
19
True
or not?
1. Let S(n)
be statements, one for each n>0.
Suppose that
(a)
the statement S(1)
is true,
(b)
for each n>1,
if S(n-1)
is true, then S(n)
is true.
Then each statement S(n) is true.
2. Let A be an nxn matrix.
Suppose that all entries in the first row of A are zero. Then det A = 0.
last page of
Lecture 18.
Lecture
20
True
or not?
1. Let A
be an arbitrary nxn matrix. Then
(a) The image of LA: Rn -> Rn is spanned by
the columns of A.
(b) rank(A) = #(linearly
independent columns of A).
2. Let {v1, v2, ..., vn} be a linearly
dependent set of vectors in a vector space V, i.e.,
a1v1 + a2v2+ ...+ anvn = 0
for some coefficinets ai (not all zero). Then for some k,
vk=b1v1 + ...+ bk-1vk-1 + bk+1vk+1 + ...+ bnvn .
3. Let A be an nxn matrix. If B is obtained from A by
(a) interchanging two rows, then det(B)=(-1) det(A)
(b) scaling a row of A by a multiple k, then det(B)=k det(A)
(c) adding a multiple of a row to
another row, then det(B)=det(A).
4. If A
contains a row of zeros, then det(A)=0.
5. If rank(A)=n, then A is a product of
elementary matricies.
Lecture
20.5
True
or not?
1. Let A be an invertible matrix. Then
det(A) det(A-1)=1.
In particular, det(A) is not zero.
2. If rank(A)<n,
then det(A)=0
(size of A is n).
3. A matrix A
is invertible if and only if det(A)
is not zero.
4. Let T: V -> W
be a linear transformation of vector spaces of dimension n..
Then there are ordered basis in V
and W such
that the matrix of T
in those bases is In.
5. Let T: V -> V be
a linear transformation of a vector space of dimension n. Then in some
ordered basis, the matrix of T
is
a1 0
... 0
0 a2
... 0
... ... ... ...
0 0 an
Lecture 22
True or not
1. A linear operator T
on a finite dimensional vector space V is diagonalizable
if and only if there is a basis of
V that consists of eigenvectors of T.
2. The multiplicity m
of an eigenvalue λ of T is the largest
positive integer for which
( t - λ
) m
is a factor of the
characteristic polynomial det(T-tI)
of T.
3. For the eigenspace E
corresponding to an eigenvalue λ of
multiplicity m,
there is an inequality:
dim (E) ≥ m.
4. The complex
conjugate of a complex number a+bi is defined to be
the complex number a-bi.
5. The absolute
number of a+bi is defined to be
|a+bi|=sqrt(a2+ b2).
Lecture 24
True or not
1. A basis β is orthonormal if any two disctinct vectors
in β are
perpendicular and every vector in β is of length 1. (careful!)
2. If {v1, v2, ..., vn} is an orthonormal basis for V and
y=a1v1 + a2v2+ ...+ anvn,
then ai
is the scalar <y,
vi>
for each i.
3. A linearly dependent set can not be orthogonal.
Lecture 26
True or not
1. An inner product on a vector space V is a map V × V -> F that satisfies
(a) <x+y, z> = <x, z> + <y, z> (b) <ax, z> = a<x, z> (c) <x, y> = <y, x>
(d) <x, x> ≥ 0 2. Let T be a linear transformation on an inner product space V. Then the adjoint T* of T is a transformation that satisfies
<T(x), y>=<x, T*(y)> for all x,y in V.
3. Let A be an nxn matrix. Then A*= At if and only if all entries of A are real numbers. 4. (T+U)*=T*+ U* 5. Let A be an nxn matrix. Then det(A)=0 if and only if rank(A)<n. 6. Every polynomial p(t) splits over complex numbers, i.e., can be written as
p(t)= (t - a1) (t - a2) ... (t - an)
for some complex numbers a1,..., an.
Lecture 27
True or not
Everywhere T is a linear transformation on an inner product space V of dimension n.
1. If λ is an eigenvalue of T, then λ is an eigenvalue of T*. 2. If x is an eigenvector of T, then x is an eigenvector of T*.3. If {v1, v2, ..., vn} is an orthonormal basis for V and
y=a1v1 + a2v2+ ...+ anvn,
then ai
is the scalar <y,
vi>
for each i. 4. If the scalars are complex numbers, then for each T there is an orthonormal basis for V such that [T] is upper triangular. 5. If T is diagonalizable, then there is an orthonormal basis for V that consists of eigenvectors of T.
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