Geoscience Reference
In-Depth Information
Reduced row echelon form of a matrix
—A matrix is in reduced row echelon form if
1. The matrix is in row echelon form.
2. The first nonzero entry in each nonzero row is the number 1.
3. The first nonzero entry in each nonzero row is the only nonzero entry in its column.
Row equivalent matrices
—Two matrices are row equivalent if one can be obtained from the
other by a sequence of elementary row operations.
Row operations
—The elementary row operations performed on a matrix are
1. Interchange two rows.
2. Multiply a row by a nonzero scalar.
3. Add a constant multiple of one row to another.
Row space of a matrix
—The subspace spanned by the rows of the matrix considered as a set
of vectors.
Similar matrices
—Matrices
A
and
B
are similar if there is a square invertible matrix
S
such
that
S
-1
AS = B
.
Singular matrix
—Square matrix
A
is singular if the equation
A
x
= 0 has a nonzero solution
for
x
.
Span of a set of vectors
—The span of the set of vectors {
v
1
, …,
v
k
} is the subspace
V
consisting
of all linear combinations of
v
1
, …,
v
k
. One also says that the subspace
V
is spanned by the
set of vectors {
v
1
, …,
v
k
} and that this set of vectors spans
V
.
Subspace
—A subset
W
of R
n
is a subspace of R
n
if
1. The zero vector is in
W
.
2.
x
+
y
is in
W
whenever
x
and
y
are in
W
.
3.
a
x
is in
W
whenever
x
is in
W
and
a
is any scalar.
Symmetric matrix
—Matrix
A
is symmetric if it equals its transpose; that is,
A
=
A
T
.
REFERENCES AND RECOMMENDED READING
Masser, M.P. and Jensen, J.W. (1991).
Calculating Area and Volume of Ponds and Tanks
, USDA Grant
89-38500-4516. Southern Regional Aquaculture Center, Washington, DC.
Stapel, E. (2013).
The Order of Operations: More Examples.
Purplemath, http://www.purplemath.com/mod-
ules/orderops2.htm.
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