Topic: Multiplication/Two digit numbers
Standard: Multiply or divide to solve word problems involving multiplicative comparison,
e.g., by using drawings and equations with a symbol for the unknown number to represent the
Systems of Linear Equations
Consider the equation
2x + y = 3.
This is an example of a linear equation in the variables x and y. As it stands, the statement
2x + y = 3 is neither true nor untrue : it is just a statement involving
Graphs of Cosecant, Secant, and Cotangent
This figure shows the Graphs of Cosecant (t), Secant (t), and Cotangent (t) on the
coordinate plane from -2 to 2 radians. Each reciprocal trig function, Csc(t), Sec(t), and
Cot(t), is graphed by using the animatio
Graphs of Sine, Cosine, and Tangent
This figure shows the Graphs of Sine, Cosine, and Tangent on the unit circle from 0 to
4 radians. Each trig function, sin(t), cos(t), and tan(t) is graphed by using the animation
or the t slider. The figure also shows t
Vector Geometry in R3
Vectors in R3
3-dimensional Euclidean space, denoted R3 , is described by three orthogonal coordinate axes
labelled X, Y and Z as shown. A point in R3 is described by by an ordered triple of real
numbers, comprising its
Transformations of Reciprocal Trigonometric Functions
This figure shows the graphs of Transformations of Reciprocal Trigonometric
Functions and their equations, in simplified form. The functions are graphed on the
coordinate plane from -2 to 2 radians. Ea
Transformations of Trig Functions
This figure shows equations of trig functions in simplified form, and graphs the
Transformations of Trig Functions on the coordinate plane from -2 to 2 radians.
Each trig transformation, in the form y = a sin[b(t + c)] +
CALCULUS FLASH CARDS
Based on those prepared by Gertrude R. Battaly (http:/www.battaly.com/), with her gracious permission.
Instructions for Using the Flash Cards:
1. Cut along the horizontal lines only.
2. Fold along the vertical lines. This will result
Logic, Sets, and
Learning Objectives for Section 7.1
The student will be able to formulate and analyze
propositions and compound propositions using
The student will be able to set up and analyze trut
Selected Solutions for 12.5
Exercise 34: Find an equation for the plane that passes through the point
(6, 0, 2) and contains the line x = 3t, y = 1 + t, z = 7 + 4t.
Solution: In order to nd the equation for a plane, we need a point in the
Worksheet 4 Solutions, Math 53
Vector Geometry and Vector Functions
Monday, September 17, 2012
1. Find an equation of the plane:
(a) The plane through the point (2, 4, 6) and parallel to the plane z = x + y.
Rewriting the original plane in the st
10. The vectors are V1 = 6.0i + 8.0j, V2 = 4.5i 5.0j.
(a) For the magnitude of V1 we have
V1 = (V1x2 + V1y2)1/2 = [( 6.0)2 + (8.0)2]1/2 =
We find the direction from
tan 1 = V1y/V1x = (8.0)/( 6.0) = 1.33.
From the signs of the components, we have
Def. 1. A set W of vectors in R n is called a subspace of R n if
it has the following three properties:
(a) The zero vector 0 belongs to W .
(b) Whenever vectors u and v belong to W , then u v belongs
to W .
(c) Whenever vector u bel
Linear Combinations, Matrix-Vector Products,
and Special Matrices
Def.1. A linear combination of vectors u , u ,., u is a sum of
scalar multiples of these vectors, that is, a vector of the form
c , c ,., c are scalars. These scalars are
c u c
Determinants. Cofactor Expansion
Def. 1. Let
be a matrix. The scalar is called the
DETERMINANT of the matrix and denoted by or . So, we
Ex. 1. Compute the determinant of the given matrix
Def. 2. We define the determinant of an mat
Properties of Determinants
We need to know how a determinant will behave if we perform a
linear operation with its rows. Row linear operations will be
used to compute any determinant without cofactor
expansion. We need to find some connection
Systems of Linear Equations
Def. 1. A system of linear equations is a set of m linear
equations in the same n variables, where m and n are positive
integer numbers. Such a system has the form
a11 x1 a12 x 2 . a1n x n b1
a 21 x1 a 22 x 2 . a 2
Linear Dependence and Linear Independence
Def. 1. A set of vectors u , u ,., u in the space R is called
linearly dependent if there exist scalars c , c , , c , not all
c u c u . c u 0 . We say also that the
zero, such that
vectors u , u , , u
Composition and Invertibility of Linear Transformations
We begin this Section 2.7 with the next useful result.
Proposition 1. The range of a linear transformation
equals the span of the columns of its standard matrix.
The Inverse of a Matrix
We start to answer the questions: (a) when a matrix has an inverse?
(b) how can we find the inverse of a matrix? Let me remind you the
definition of the inverse of a matrix first (see on the board).
Theorem 2.5. Let A b
Invertibility and Elementary Matrices
Def. 1. An n n matrix A is called invertible if there is an
n n matrix B such that A B I , B A I . At this case matrix B
is called the inverse of A and denoted by A .
Ex. 1. Determine whether B A for the g
Linear Transformations and Matrices
Def. 1. Let S1 and S 2 be subsets of R n and R m , respectively. A
function f from S1 into S 2 , written f : S1 S 2 , is a rule that
assigns to each vector x S1 a unique vector f ( x) S 2 . The vector
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