A zero matrix is a matrix all of whose entries are 0. If you add the
matrix to another
matrix A, you get A:
In symbols, if
zero
is a zero matrix and A is a matrix of the same size, then
A zero matrix is said to be an identity element for matrix addition.
If A is a matrix, the transpose
columns of A. For example,
Notice that the transpose of an
of A is obtained by swapping the rows and
matrix is an
matrix.
Example. Consider the following matrices with real entries:
(a) Compute
.
(b) Compute
.
Example. The
Linear Algebra I Final Examination. Due May 3 2011 5pm
This is a take-home examination. Do not share solutions with other students. Show all working
in sucient detail that it is clear that all logical issues have been resolved.
(1) (9 points) Suppose that
Corollary.
is a field if and only if n is prime.
Example. ( Fields of prime characteristic)
,
, and
are fields, since 2, 3,
and 61 are prime.
On the other hand,
is not a field, since 6 isn't prime (because
).
is a
commutative ring with identity.
For simpl
The integers mod n is the set
n is called the modulus.
For example,
becomes a commutative ring with identity under the operations of addition mod
n and multiplication mod n. I'm going to describe these operations in a functional
way, which is sufficient f
Write the system of linear equations
as a matrix multiplication equation.
Example. Write the system of equations which correspond to the matrix equation
Multiply out the left side:
Equate corresponding entries:
Example. ( Identity matrices) There are spec
A matrix is a finite rectangular array of numbers:
In this case, the numbers are elements of (or ). In general, the entries will be
elements of some commutative ring or field.
I'll explain operations with matrices in the following examples. I'll discuss a
Different algebraic systems are used in linear algebra. The most important
are commutative rings with identity and fields.
Definition. A ring is a set R with two binary operations addition (denoted +)
and multiplication (denoted ). These operations satisf
Linear Algebra I Assignment 3. Due April 19 2011
1. Let M be a n n matrix and suppose that
M=
I
0
B
A
where A is a j j matrix with j < n; I is the (n j ) (n j ) identity
matrix and 0 a matriz of zeros. Prove that det(M ) = det(A).
2. Let A and B be two sq
Linear Algebra I Assignment 2. Due April 5 2011
1. Consider linear maps from a vector space X onto itself. Consider the map
O given by
x X
O(x) = 0
Show that O is linear. Now suppose we have another map A such that
AA=O
Show that A has no inverse map.
2.
Linear Algebra I Assignment 1. Due March 1 2001
1. Suppose there is a three dimensional vector space V and the vectors u, v, w
form a basis. Show that the vectors u + v + w, v + w and w also form a
basis.
2. Suppose that W1 and W2 are nite dimensional sub