Math 200 Spring 2015
February 18
a b
Definition. Let A =
. The determinant of A, denoted det(A)
c d
or |A|, is given by
det(A) = ad bc.
Theorem. Let A be a 2x2 matrix. The equation A~v = w
~ has a
2
solution ~v for every vector w
~ in R if and only if det
Math 200 Spring 2015
February 16
Definition. A matrix is a rectangular array of numbers. An m by n
matrix has m rows and n columns.
a b
x
Definition. For the matrix A =
and the vector ~v =
,
c d
y
ax + by
the product A~v =
.
cx + dy
3 4
2
Example. For
Math 200 Spring 2015
February 27
Notice that in R2 , if ~v is a scalar multiple of w,
~ say ~v = k w,
~ where k 6= 0,
then we can rearrange the equation and write (1)~v + (k)w
~ = ~0. In words,
we can write the zero vector as a linear combination of ~v an
Math 200 Spring 2015
February 20
Definition. A linear transformation from R2 to R2 is a function
T : R2 R2 such that, for all ~v , w
~ R2 and all k R,
T (~v + w)
~ = T (~v ) + T (w)
~ and
T (k~v ) = kT (~v )
a11 a12
Definition of matrix multiplication:
Math 200 Spring 2015
February 23
Definition. The set B = cfw_~v1 , ~v2 is a basis for R2 if every vector in R2
can be written uniquely as a linear combination of the vectors ~v1 and ~v2 .
Examples.
The set cfw_~e1 , ~e2 is a basis for R2 ; its called t
Math 200 Spring 2015
February 9
Definition. A vector in R2is an
ordered pair ~v of real numbers. It can be
a
expressed in column form
or row form a b . The zero vector
b
~0 = 0 0 .
We visualize a vector in in R2 as a directed line segment an arrow
start
Math 200 Spring 2015
February 25
Moving up to R3 .
We define R3 in analogy with R2 ; that is, R3 is the set of ordered triples
of real numbers. We associate to each such triple (a, b, c) the directed line
segment from
the origin to the point (a, b, c) in
Math 200 Spring 2015
February 11 (Revised version)
Definition. A linear combination of the vectors ~v1 , ~v2 , ., ~vn is a vector
of the form a1~v1 + a2~v2 + . + an~vn , where a1 , a2 , ., an are scalars.
Example. In problems
3 - 5 of the February 13 home