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66 Cards in this Set

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What is a vector?
A vector is a directed line segment that corresponds to a displacement from one ponit A to another point B.
Intitial point is also called the _______.
tail
Terminal point is also called the ________.
head
What are the numbers inside a vector called?
components
What is the one vector that you cannot really draw in any dimention? What is it denoted by?
The zero vector cannot really be drawn in any dimention as it has not length. It is denoted as 0.
What does equal mean when referencing vectors?
Equal in the context of vectors means two vectors of the same length and the same direction.
What can be said of a vector when it is in standard position?
It can be said that the vector's tail is positioned at (0,0).
How do you perform vector addition given the two vectors u and v?
The two vectors must have the same dimentions. For every a m x n matrix add each piece (every m and n combination) of the vectors together to create a new vector.
What is the Head to Tail Rule
Given Vector u and v in "r two" translate v so that its tail coincides with the head of u. The sum u + v of u an dv is the vector from the tail of u to the head of v.
What is the parallelogram rule?
Given vectors u and v in "r two" (in standard position), their sum u + v is the vector in standard position along the diagonal of the parallelogram determined by u and v.
how do you perform scalar multiplication given a scalar c and a vector v?
cv = c[v₁, v₂] = [cv₁, cv₂]
Where does the word vector come from?
The word vector comes from the Latin root meaning "to carry". A vector is formed when a point is displaced---or "carried off"---a given distance in a given direction.
Where does the word component come from?
The word component is derived from the Latin words co, meaning "together with," and ponere, meaning "to put." Thus, a vector is "put together" out of its components.
Where does the word scalar come from?
The term scalar comes from the Latin word scala, meaning "ladder." The equally spaced rungs on a ladder suggest a scale, and in vector arithmetic, multiplication by a constant changes only the scale (or length) of a vector. Thus, constants became known as scalars.
Where does the word theorem come from?
The word theorem is derived from the greek word theorema, which in turn comes from a word meaning "to look at." Thus, a theorem is based on the insights we have when we look at examples and extract from them properties that we try to prove hold in general.
What are the algebraic properties of vectors? How many are there?
Let u, v, and w be vectors in "r to the n", let c and d be scalars. Then

a. u + v = v + u Communtativity
b. (u + v) + w = u + (v + w) Asociativity
c. u + 0 = u
d. u + (-u) = 0
e. c(u + v) = cu + cv Distributivity
f. (c + d)u = cu + du Distributivity
g. c(du) = (cd)u
h. 1u = u
What is a linear combination?
A vector v is a linear combination of vectors v₁, v₂, ...., "v of n" if there are scalars c₁, c₂, ...., "c of n" such that
v = c₁v₁, c₂v₂, ...., "c of n" times "v of n". The scalars in the linear combination are called the coeficients.
How do you perform the dot product? Assume you are given vectors u and v.
The dot product is performed by taking each corresponding component of u and v and multiplying them together. Each of these multiplied components is then added together. The result of this is that a dot product generates a scalar out of two vectors.

u "dot product" v = v = u₁v₁ + u₂v₂ + .... + "c of n" times "v of n"
What are the properties of the dot product? How many of them are there?
Let u, v, and w be vectors in Rⁿ and let c be a scalar. Then

a. u "dot product" v = v "dot product" u Commutativity
b. u "dot product" (v + w) = u "dot product" v + u "dot product" w
c. (cu) "dot product" v = c(u "dot product" v)
d. u "dot product" v is greater than or equal to zero. (and u "dot product" u = 0 iff u = 0)
What is the length of a vector? What is the length also called?
The length (or norm) of a vector v in Rⁿ is the nonnegative scalar ||v|| defined by
||v|| = sqrt(v "dot product" v)

In words, the length of a vector is the square root of the sum of the squares of its components.
What is a vector of length 1 called? What is the process of transforming a vector into a vector of length 1 called?
A unit vector. It is called normalization.
What is the Cauchy-Schwartz Inequality?
For all vectors u and v in Rⁿ,

|u "dot product" v| is less than or equal to ||u|| * ||v||
What is the triangle inequality
For all vectors u and v in Rⁿ,

||u + v|| is less than or equal to ||u|| + ||v||

This is simple to realize when you consider that it basically says that the length of u + v is always greater than the length of u plus the length of v
How do you compute the distance between two vectors u and v?
The distance between two vectors u and v in Rⁿ is defined by

d(u, v) = ||u - v||
What formula can be used to take two vectors and get a number that is equal to the cos(theta)?
For nonzero vectors u and v in Rⁿ,

cos(theta) = u "dot product" v / ||u|| * ||v||
Give the cosine of all the special angles?
The cosine of special angles are as follows

degree 0 = sqrt(4)/2 = 1
degree 30 = sqrt(3)/2
degree 45 = sqrt(2)/2 = 1/sqrt(2)
degree 60 = sqrt(1)/2 = 1/2
degree 90 = sqrt(0)/2 = 0
What does orthogonal mean and how can it be determined for two vectors?
Two vectors u and v in Rⁿ are orthogonal to each other if u "dot product" v = 0
What is Pythagoras Theorem?
For all vectors u and v in Rⁿ, squared(||u + v||) = squared(||u||) + squared(||v||) if and only if u and v are orthogonal.
How do you find the projection of vector v onto vector u? What is a good visual for the projection?
If u and v are vectors in Rⁿ and u is not equal to 0, then the projection of v onto u is,

proj(v) onto u = (u "dot product" v / u "dot product" u) times u.

A good visual for a projection is to parralel rays of light that are comming at u and that are prependicular to u. The projection of v onto u is the shadow cast by v where the parralel beams of light do not reach u because they are blocked by v.
What is the word orthogonal derived from?
The word orthogonal is derived fromt he Greek word orthos, meaning "upright," and gonia meaning "angle." Hence, orthogonal literally means "right-angled." The latin equivilant is rectangular.
Where does the word trivial come from?
The word trivial is derived from the latin root tri-("three") and the Latin word via ("road"). Thus, speaking litterally, a triviality is a place where three roads meet.
What is the trivium?
The trivium consisted of the three common subjects: grammar, rhetoric, logic that were taught before the quadrivium (arithmetic, geometry, music, and astronomy).
What is a linear equation in n variables?
A linear equation in n variables x₁, x₂, ..., "x to the n" is an equation that can be written in the form

a₁x₁ + a₂x₂, ..., "a to the n" times "x to the n" = b

where the coefficientes a₁, a₂, "a to the n" and the constant term b are constant.
What is a system of linear equations?
A system of linear equations is a finite set of linear equations, each with the same variables.
What is a solution of a system of linear equations? What is this also called?
A solution of a system of linear equations is a vector that is simultaneously a solution of each equationin the system. This is also called "solving the system".
Contrast Consistent vs. Inconsistent linear equations?
A system of linear equations is called consistent if it has at least one solution. A system with no solution is called inconsistent.
Two linear system are called ______ if they have the same solution sets.
equivalent
What is back substitution? How do you do back substitution?
Back substitution is a method for solving a system of linear equations. It involves solving first for the linear equations that you can currently solve for and then using the solved for equations to solve for other equations. Systems that are arranged in back substitution ways have a variable at every row and are solved bottom up.
Where does the word matrix come from?
The word matrix is derived from the Latin word mater, meaning "mother." When the suffix -ix is added, the meaning becomes "womb." Just as a womb surrounds a fetux, the brackets of a matrix suround its entries, and just as the womb gives rise to a baby, a matrix gives rise to certain types of functions called linear transformations.
What is an augmented matrix?
A augmented matrix is a matrix that encompasses the system of linear equations coefficients and the thing the linear equations are equal to. The thing they are equal to goes in the rightmost column. A line is often drawn between the right column and the left column to remind us of the equal sign. The augmented matrix is just the coefficient matrix with an extra constant containing the constant terms.
What does ill-conditioned mean in the context of mathematics?
Ill-conditioned systems are systems that are extremely sensitive to roundoff error, and there is not much we can do about it.
What is a coefficient matrix?
A coeficient matrix is simply the direct substituion of the system of linear equations into matrix form. It is formed from the coefficients of the linear equations in the systems.
Where does the word echelon come from?
The word echelon comes from the Latin word scala, meaning "ladder" or "stairs." the French word for "ladder," echelle, is also derived from this Latin base. A matrix in echelon form exhibits a staircase pattern.
What is row echelon form?
A matrix is in row echelon ofrm if it satisfies the following properties:
1) Any rows consisting entirely of zeros are at the bottom
2) In each nonzero row, the first nonzero entry (called the leading entry) is in a column to the left of any leading entries below it. (basically this guarantees that the leading entries form a staicase pattern)
What elementary row operations can be performed on a matrix?
The following elementary row operations can be performed on a matrix:

1) Interchange two rows
2) Muliply a row by a nonzero constant
3) Add a multiple of a row to another row.
What is row reduction?
The process of applying elementary row operations to bring a matrix into row echelon form is called row reduction.
Is the row echelon form of a matrix unique?
No, the row echelon form of a matrix is not unique, many matrix have several possible row echelon forms.
Are elementary row operations reversible?
Elementary row operations are reversible-- that is, they can be "undone." Thus, if some elementary row operation converts A into B, there is a also an elementary row operation that converts B into A.
What is row equivalent?
Matrices A and B are row equivilant if there is a sequence of elementary row operations that converts A into B.
Matrices A and B are ________________ iff they can be reduced to the same row echelon form.
row equivalent
What is Gaussian elimination?
When row reduction is applied to the augmented matrix of a system of linear equations to create a equivalent system that can be solved by back substitution the entire process is known as Gausian elimination
What is the rank of a matrix?
The rank of a matrix is the number of nonzero rows in its row echelon form. Denoted by

rank(A)
What is the Rank Theorem?
Let A be the coefficient matrix of a system of linear equations with n variables. If the ssystem is consistent, then

number of free variables = n - rank(A)
What is reduced row echelon form?
A matrix is in reduced row echelon form if it satisfies the following properties:

1) It is in row echelon form
2) The leading entry in each nonzero row is a 1 ( called a leading 1).
3) Each column containing a leading 1 has zeros everywhere else.
Is reduced row echelon form unique?
Yes! The reduced row echelon form a of a matrix is unique.
How do you perform Gauss-Jordan elimination?
To perform Gauss-Jordan elimination:

1) Write the augmented matrix of the system of linear equations.
2) Use elentary row operations to reduce the augmented matrix to reduced row echelon form
3) If the resulting system is consistent, solve for the leading variables in terms of any remaining free variables.
A system of linear equation sis called __________ if th econstatn form in each equation is zero.
homogenous
What is a span?
If S = {v₁, v₂, ..., "to v of k"} is a set of vectors in Rⁿ, then the set of all linear combinations of v₁, v₂, ..., "to v of k" is called the span of v₁, v₂, ..., "to v of k" and is denoted by span(v₁, v₂, ..., "to v of k")or span(S).
What is a spanning set?
If span(S) = Rⁿ, then S is called a spanning set for Rⁿ.
What does it mean to say that a set of vectors is linearly dependent?
We say that a set of vectors are lineary independent if one of them can be written as a linear combination of the others. Or Formally:

A set of vectors v₁, v₂, ..., "v of k" is linearly dependent if there are scalars c₁, c₂, ..., "c of k" at leaast one of which is not zero, such that

c₁v₁ + c₂v₂ + ... + "c of k" * "v of k"
What does it mean to say that a set of vectors is linearly independent?
Means that it is not linearly dependent.
Given a set of m vectors in Rⁿ, if m > n, what does this imply?
It implies that the m vectors are linearly dependent.
Vectors v₁, v₂, ..., "v of k" in Rⁿ are linearly dependent _______FINISH THIS STATEMENT________?
Vectors v₁, v₂, ..., "v of k" in Rⁿ are linearly dependent if and only if at least one of the vectors can be expressed as a linear combination of the others.
What is the rank vector theorem?
If v₁, v₂, ..., "v of k" be row vectors in Rⁿ. Then if a m X n matrix is created of all these vectors as its rows v₁, v₂, ..., "v of k" are linearly dependent if and only if rank(A) < m.
What is Ohm's Law?
force = resistance * current

E = RI
What are Kirchhoff's Laws?
Current Law (nodes):
The sum of the currents flowing into a ny node is equal to the sum of the current flowing out of that node.

Voltage Law (circuits):
The sum of the voltage drops around any circuit is equal to the total voltage around the ciruit (provided to the batteries)