midterm 1 (week 0-4)

Question 1

intersection at 1 point

Answer 1

unique x, y, z values

Question 2

intersection on a line

Answer 2

having a variable determined by another variable
ex. x = z + 2
y = -2z -1

Question 3

no intersection

Answer 3

a system is inconsistent
ex. 0 = -6

Question 4

vector equation of a line form

Answer 4

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

Question 5

set notations

Answer 5

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

Question 6

reading set notation

Answer 6

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”

Question 7

subset

Answer 7

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

Question 8

equality of sets

Answer 8

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.

Question 9

union of sets

Answer 9

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}

Question 9

intersection of sets

Answer 9

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}.

Question 10

column vector

Answer 10

a matrix with only one column and multiple rows

Question 11

row vector

Answer 11

a matrix with only one row and multiple columns

Question 12

adding matrices

Answer 12

v + w = v1 + w1 , v2 + w2 … (actually add the values to get new ones).

Question 13

multiplying by a scalar

Answer 13

kv = k(v1, v2…) = (kv1, kv2…)

Question 13

PQ

Answer 13

Q - P
q1-p1, q2-p2

Question 14

length or norm of a vector

Answer 14

II x II = sqrt of x1^2 + x2^2 …

Question 15

dot product

Answer 15

v . w = v1w1 + v2w2…
also v . w = cosθ IIvII IIwII

Question 16

angle between vectors

Answer 16

cos-1 (v . w / II v II II w II)

Question 17

perpendicular or orthogonal vectors

Answer 17

v . w = 0

Question 18

parametric form

Answer 18

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

Question 19

normal vector

Answer 19

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}

Question 20

coefficient matrix

Answer 20

the numbers multiplied by xyz or x1x2x3 (the numbers when you drop the variables but keep them in your mind)

Question 21

augmented matrix

Answer 21

has a line separating the matrix structure and scalar solution

Question 22

elementary row operations

Answer 22

1. interchanging two rows
2. multiplying one row by a nonzero number
3. adding or subtracting a row from another row`

Question 23

REF

Answer 23

1. all zero rows are at the bottom
2. 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
3. all entries below a leading entry are zero

there are many REFs of a matrix

Question 24

RREF

Answer 24

1. must satisfy REF rules
2. all leading entries are 1 (leading 1s)
3. each leading 1 is the only nonzero entry in its column

there is only one RREF of a matrix

Question 25

rank of a matrix

Answer 25

the number of leading ones in RREF

Question 26

pivot position

Answer 26

a pivot position in a matrix B is a location in B that corresponds to a leading one in the RREF

Question 27

pivot column

Answer 27

a column of B that contains a pivot position

Question 28

free variable

Answer 28

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)

Question 29

solving a system of linear equations

Answer 29

1. write the system in augmented matrix form
2. row reduce to RREF
3. identify free variables and basic variables
4. put RREF into equation form
5. solve for basic variables in terms of free variables
6. set a parameter for the free variables and write the equations

Question 30

solution type (theorem)

Answer 30

consistent = at least one solution
- at least one free variable = infinitely many solutions
- all leading variables = exactly one solution

inconsistent = no solution
- 0 = 1

Question 31

matrix-vector product

Answer 31

multiplying a matrix by a vector to get Ax

Question 32

algebraic rules for Ax (theorem)

Answer 32

1. A(x + y) = Ax + Ay
2. A(kx) = k(Ax)

Question 33

matrix form of a linear system (theorem)

Answer 33

[A|b] in matrix form as Ax = b

Question 34

the product of Ax in terms of columns (theorem)

Answer 34

Ax = (v1 v2 v3…)(x1 x2 x3…) = x1v1 + x2v2…

where v a row vector and x is a column vector

Question 35

transformations or maps

Answer 35

another word for function and can be used interchangeably

take inputs in their domain (R^m) and have outputs in their codomain (R^n)

Question 36

linear transformation rules

Answer 36

a transformation T : R^m –> R^n is a linear transformation if (and only if)

1. T(u + v) = T(u) + T(v)
2. T(ku) = kT(u)

note that if u and v are parallel vectors so is T(u) and T(v)

Question 37

how do you know if a transformation is a linear transformation?

Answer 37

apply the rules/theorem and see if it applies

Question 38

standard vector

Answer 38

a unit vector with zeros in all entries except for a 1 in ith entry

Question 39

rotation matrix

Answer 39

[cos θ −sin θ]
[sin θ cos θ]

Question 40

matrix algebra

Answer 40

associativity: (AB)C = A(BC)
distributivity: C(A+B) = CA + CB
scalar multiplication: k(AB) = A (kB) = (AB)k

Question 41

injective

Answer 41

no two x’s (inputs) map to the same y’s (outputs)

Question 42

surjective

Answer 42

for every input there is an output (they can share)

Question 43

bijective

Answer 43

each unique input has a unique output. if a matrix is bijective, it is invertible/has an inverse

Question 44

criteria for invertibility (theorem)

Answer 44

if A is a n x n matrix, and the RREF of A = In then A is invertible

Question 45

inverting a matrix (theorem)

Answer 45

Inverse of A can be computed by A=In and performing elementary row operations until A reaches RREF resulting in In=Ainverse