midterm 1 (week 0-4) Flashcards
intersection at 1 point
unique x, y, z values
intersection on a line
having a variable determined by another variable
ex. x = z + 2
y = -2z -1
no intersection
a system is inconsistent
ex. 0 = -6
vector equation of a line form
making a variable equal an arbitrary number (t) to form a solution
ex. z = t. x = t + 2, y = -2t -1
(x, y, z) = (t + 2, -2t - 1, t)
= (2, -1, 0) + t(1, -2, 1)
vector equation of a line
set notations
R - the set of all real numbers
Z - the set of all integers or whole numbers
Q - the set of all rational numbers
N - the set of all natural numbers
C - the set of all complex numbers
{ } or o with dash - empty set
reading set notation
P = {an expression describing a typical element in P I specifying the parameters used in the description}
ex. “P is the set of all s in R such that s is even”
subset
we say set A is a subset of a set B if all the elements of A are also in B
A⊆B, if for every a ∈A, a ∈B
equality of sets
we say sets A and B are equal if A is a subset of B and B is a subset of A
A = B if A ⊆B and B ⊆A.
union of sets
a set that contains all elements of A and B
A ∪B = {x ∈X |x ∈A or x ∈B}
ex. Let A = {2,5,7,π} and B = {4,π,5} be subsets of R. Then A ∪B = {2,5,7,π,4}
intersection of sets
a set that contains all common elements between A and B
A ∩B = {x ∈X |x ∈A and x ∈B}
ex. Let A = {2,5,7,π} and B = {4,π,5} be subsets of R. Then A ∩B = {5,π}.
column vector
a matrix with only one column and multiple rows
row vector
a matrix with only one row and multiple columns
adding matrices
v + w = v1 + w1 , v2 + w2 … (actually add the values to get new ones).
multiplying by a scalar
kv = k(v1, v2…) = (kv1, kv2…)
PQ
Q - P
q1-p1, q2-p2
length or norm of a vector
II x II = sqrt of x1^2 + x2^2 …
dot product
v . w = v1w1 + v2w2…
also v . w = cosθ IIvII IIwII
angle between vectors
cos-1 (v . w / II v II II w II)
perpendicular or orthogonal vectors
v . w = 0
parametric form
x1 = p1 + kd1
x2 = p2 + kd2
where p is a point, k is a scalar and d is the direction vector
ex. r = (2, 0, 4) + k(1, -3, 0)
x1 = 2 + t
x2 = -3t
x3 = 4
normal vector
State the normal vector and then state l is the set of vectors in Rn perpendicular to the normal
l = {x ∈R2 |x ·n = 0}
coefficient matrix
the numbers multiplied by xyz or x1x2x3 (the numbers when you drop the variables but keep them in your mind)
augmented matrix
has a line separating the matrix structure and scalar solution
elementary row operations
- interchanging two rows
- multiplying one row by a nonzero number
- adding or subtracting a row from another row`
REF
- all zero rows are at the bottom
- the leading entry in each row is to the right of the leading entry of the row above (staircase) - doesn’t have to be ones
- all entries below a leading entry are zero
there are many REFs of a matrix
RREF
- must satisfy REF rules
- all leading entries are 1 (leading 1s)
- each leading 1 is the only nonzero entry in its column
there is only one RREF of a matrix
rank of a matrix
the number of leading ones in RREF
pivot position
a pivot position in a matrix B is a location in B that corresponds to a leading one in the RREF
pivot column
a column of B that contains a pivot position
free variable
a column that is not a pivot column (has no leading one)
means a system has infinite solutions (because you can give it any value and change the values of other variables)
solving a system of linear equations
- write the system in augmented matrix form
- row reduce to RREF
- identify free variables and basic variables
- put RREF into equation form
- solve for basic variables in terms of free variables
- set a parameter for the free variables and write the equations
solution type (theorem)
consistent = at least one solution
- at least one free variable = infinitely many solutions
- all leading variables = exactly one solution
inconsistent = no solution
- 0 = 1
matrix-vector product
multiplying a matrix by a vector to get Ax
algebraic rules for Ax (theorem)
- A(x + y) = Ax + Ay
- A(kx) = k(Ax)
matrix form of a linear system (theorem)
[A|b] in matrix form as Ax = b
the product of Ax in terms of columns (theorem)
Ax = (v1 v2 v3…)(x1 x2 x3…) = x1v1 + x2v2…
where v a row vector and x is a column vector
transformations or maps
another word for function and can be used interchangeably
take inputs in their domain (R^m) and have outputs in their codomain (R^n)
linear transformation rules
a transformation T : R^m –> R^n is a linear transformation if (and only if)
- T(u + v) = T(u) + T(v)
- T(ku) = kT(u)
note that if u and v are parallel vectors so is T(u) and T(v)
how do you know if a transformation is a linear transformation?
apply the rules/theorem and see if it applies
standard vector
a unit vector with zeros in all entries except for a 1 in ith entry
rotation matrix
[cos θ −sin θ]
[sin θ cos θ]
matrix algebra
associativity: (AB)C = A(BC)
distributivity: C(A+B) = CA + CB
scalar multiplication: k(AB) = A (kB) = (AB)k
injective
no two x’s (inputs) map to the same y’s (outputs)
surjective
for every input there is an output (they can share)
bijective
each unique input has a unique output. if a matrix is bijective, it is invertible/has an inverse
criteria for invertibility (theorem)
if A is a n x n matrix, and the RREF of A = In then A is invertible
inverting a matrix (theorem)
Inverse of A can be computed by A=In and performing elementary row operations until A reaches RREF resulting in In=Ainverse
