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Linear Algebra 1554 > Midterm 1 > Flashcards

Midterm 1 Flashcards

1
Q

Linear Equation

A

○ a1x1 + a2x2 + … + anxn = b
organization of coefficients and variables with a solution ‘b’

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2
Q

System of linear equations

A

collection of multiple linear equations

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3
Q

solution of a system

A

list of numbers that make each equation true when values are substituted in for the corresponding ‘x’ variables

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4
Q

solution set

A

set of all possible solutions of a linear system

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5
Q

equivalent linear systems

A

systems with the same solution set

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6
Q

Consistent System

A

there is at least one solution (1 or infinite many)

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7
Q

Inconsistent System

A

there is no solution to the system for a specific input

they have EMPTY solution sets

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8
Q

existence and uniqueness questions

A

existence: does the solution/something exist?
uniqueness: if the solution exists, is there more than one solution (infinite many)?

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9
Q

How many solutions can a linear system have

A

none, one, or infinite many

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10
Q

size of a matrix

A

m x n
m: number of rows
n: number of columns

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11
Q

Row reduction operations

A
  1. take a multiple of one row and ADD that to another (used to eliminate entries)
  2. scale a row by a scalar not equal to zero (usually done to make a leading entry = 1)
  3. interchange rows
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12
Q

non-zero row/column

A

at least ONE of the entries has to be nonzero

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13
Q

leading entry

A

leftmost nonzero entry in a row

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14
Q

Conditions of Echelon Form

A
  1. leading entries of the next row are right of the leading entries in previous row above it
  2. everything in the column below a leading entry is all 0
  3. all zero rows are at the bottom
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15
Q

Row Reduced Echelon Form and Conditions!

A

represents a potential solution set for a linear system
EACH MATRIX ONLY HAS ONE RREF - row equivalent to just ONE

  1. all leading entries are 1s
  2. there are 0s ABOVE and BELOW each leading 1

if matrix is not echelon or RREF that means MORE row reductions must be done

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16
Q

pivot position and pivot columns

A

pivots are the locations in a matrix that correspond with leading 1s of RREF
pivot columns are the columns that have pivot positions

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17
Q

Basic/Leading Variables

A

variables that correspond with a pivot
basic variables have an EXACT value for a solution set

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18
Q

Free variables

A

don’t correspond to any pivots or pivot columns
can be assigned ANY value for a consistent linear system

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19
Q

Overdetermined System

A

More rows than columns
more equations than variables
- can be consistent
- can have a unique solution
– doesn’t necessarily have to be?

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20
Q

Underdetermined System

A

more columns than rows
more variables than equations
there will always be a free variable SO cannot have a unique solution
if consistent: infinite!
if inconsistent: no solution

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21
Q

Existence and Uniqueness Theorem

A

a linear system is consistent if and only if there is NOT a pivot in the last augmented column (rightmost column of augmented matrix)
● [0 0 0 0 0 | b] with b non-zero
if linear system is consistent:
1 solution (no free variables)
infinite (at least 1 free variable)

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22
Q

R^n

A

dimension we are in
n is the number of rows or entries in a vector
R is the collection of all lists if n real numbers

example:

○ R2 vector: 2 rows
○ R3 vector: 3 rows

vectors in R2 are a line to a point in 2D space
vectors in R3 are a line to a point in 3D space

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23
Q

Linear Combinations

A

Linear combinations is y = c1v1 + c2v2 + … + cpvp
where vs are a set of vectors in R^n and cs are weights or SCALARS
Y is a linear combination!

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24
Q

Span{v1 …vp}

A

Span is the set of all possible linear combinations of those sets of vectors
c1v1 + c2v2 + … + cpvp

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25
Q

Algebraic Properties of Rn (8 total)

A

○ For all u, v, w in Rn and all scalars c & d:
i. u + v = v + u v. c (u + v) = cu + cv
ii. (u + v) + w = u + (v + w) vi. (c + d) u = cu + du
iii. u + 0 = 0 + u = u vii. c (du) = (cd) u
iv. u + (-u) = -u + u = 0 viii. 1u = u

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26
Q

VECTOR EQUATIONS

A

vector equation: x1a1 + x2a2 + … + xnan = b where a and b are vectors
- has the same solution set as the linear system whose augmented matrix is [a1 a2 .. an | b]
- b can only be generated if there exists a solutions (weights ‘x’s) to the linear system with the matrix!

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27
Q

More about SPAN and connection to vector equation

A

If v1….vp are in Rn, then the set of all linear combinations of v1…vp is denoted by Span{v1…vp} this is called the subset of Rn spanned by v1….vp

is vector b in Span{v1…vp} is the same as asking x1v1+….+xpvp = b??
- solve the augmented matrix!
- is every b of Rn a linear combination of the vectors in {v1…vp}? == is there a pivot in every row

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28
Q

Identity Matrix

A

output the same input

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29
Q

● x ∈ Rn

A

“x is a vector with n elements”

30
Q

Star Equation

A

matrix A of vectors times a mx1 matrix of weights
Ax =x1a1 + x2a2 + … xnan
the number of columns of A = number of entries in x

31
Q

If A is an mxn matrix, with columns a1…an and if b is in Rm: CONNECT THE DIFFERENT FORMS

A

matrix equation = vector equation = augmented matrix for linear system
○ (Ax = b) == (x1a1 + x2a2 + … + xnan = b) == ([a1 a2 … an | b])

32
Q

Span as a relation to VECTORS

A

Ax = b as a linear combination is basically the A vector times the x vector

Span of the columns is MULTIPLYING these two parts? (basically how do you get the linear combinations)
STAR EQUATION
span = all the possible linear combinations of vectors
x1a1 + x2a2 + … + xnan = b

33
Q

Ax = b has a solution IF AND ONLY IF

A

b is a linear combination of the columns of A

34
Q

LOGICALLY EQUIVALENT STATEMENTS (4 total)

A

-For each b in Rm, the equation Ax = b has a solution
- each b in Rm is a linear combination of the columns of A
- the columns of A span Rm
-A has a pivot position in every row

35
Q

Properties of MATRIX equations

A

A(u +v) = Au +Av
A(cu) = c(Au)

36
Q

Homogeneous linear system/Trvial vs nontrivial solution/Nonhomogenous linear system

A

Ax = 0
Trivial Solution: x = 0
NonTrivial Solution: x that is a nonzero vector that satisfies Ax = 0 == AT LEAST ONE nonzero element
nonhomogenous: Ax = b where b is not 0

37
Q

Homogenous System

A

-ALWAYS ha at least one solution, always consistent, always has the trivial solution
-IF THERE is at least one free variable, homogeneous has a NONtrivial solution

38
Q

Implicit vs Explicit description of a plane

A

Implicit: ○ 10x1 - 3x2 - 2x3 = 0
Explicit is PARAMETRIC VECTOR EQUATION
○ x = su + tv === ○ x = x2u + x3v
where x is the x vector and s,t are in R
free variables!

39
Q

Parametric Vector FORMS

A

Consistent Ax = b
x = u + tv
Ax = 0
x = tv

For a consistent Ax = b, the solution set of Ax = b is the set of all vectors of the form w = p + vk where p is a solution and vk is any solution of Ax = 0

40
Q

How to Write a solution set in Parametric Vector Form

A
  1. Row reduce to RREF
  2. express each basic variable in terms of any free variable
  3. write a typical solution x as a vector whose entries depend on the free variables
  4. decompose x into a linear combination of vectors using the free variables as parameters
41
Q

Linearly Independent

A

Matrix A has a pivot in every column
NO free variables
x1v1 + x2v2 + … + xpvp = 0 HAS ONLY the trivial solution

42
Q

Linearly dependent

A

Vector equation c1v1 + c2v2 + … + cpvp = 0 where not all the weights are not all zero, meaning AT LEAST ONE nonzero weight
at least ONE free variable

43
Q

More columns than rows

A

Linearly dependent because there is going to be a free variable

likewise, IF a set of vectors is linearly independent, then number of rows will equal or be greater than the number of columns; this does not mean that if there are more rows then columns then a set of vectors is AUTOMATICALLY linearly independent

44
Q

Sets with ONE vector is linearly independent

A

IF AND ONLY IF IT IS NOT THE ZERO VECTOR

45
Q

sets with 2 or more vectors are linearly dependent IF

A

one of the vectors is a multiple of another OR one of the vectors is a linear combination OR in the span of at least two other vectors OR at least one of the vectors is a ZERO vector!!

46
Q

Matrix Transformation

A

assigns a vector x in Rn to a vector T(x) in Rm
using A to turn x into T(x)

47
Q

Linear Transformation

A

all matrix transformations are linear transformations
NOT ALL linear transformations are matrix transformations
A MATRIX TRANSFORMATION must preserve the operations of vector addition AND scalar multiplication
T(cu +dv) = cT(u) + dT(v)

48
Q

Domain/CoDomain/Image of x under the action of T/Range of T

A

Domain: input: set Rn; number of entries in x
Codomain: output: set Rm; number of entries in T(x) - same number of entries in A or number of rows of A
Image of x under the action of T: vector T(x)
Range of T: set of all possible images T(x); span of columns of A??
Principle of superposition: ○ T(c1v1 + … + ckvk) = c1Tv1 + … + ckTvk

49
Q

Solving Ax = b in the context of Transformations!

A

Trying to solve for all vectors x in Rn that are transformed into vector b in Rm under the action of multiplication by A

50
Q

T(x) = rx

A

Contraction: 0 <= r < 1
Dilation: r >1

51
Q

Standard Matrix for a linear transformation T

A

○ A = [T(e1) + … + T(en)]
e1 in R2 is (1 0)
e2 is R2 is (0 1)

52
Q

ONTO

A

existence question!!
T: Rn -> Rm is ONTO Rm if each b in Rm is the image of AT LEAST one x in Rn
Ax = b is ALWAYS consistent
at least one solution
- standard matrix has a pivot in EVERY ROW
- columns of A SPANS Rm = means every b in Rm is going to have a solution, every b is a linear combination of the columns of A

Geometric: get to any vector with an image

53
Q

ONE-TO-ONE

A

UNIQUENESS QUESTION
T: Rn -> Rm is ONE TO ONE if each b in Rm is the image of AT MOST one (possible 0) x in Rn
T(x) = Ax = b has either ONE solution or NO solution
pivot in EVERY column : no free variables!!
columns of A are LINEARLY INDEPENDENT
ONE TO ONE if and only if solution to T(x) = 0 is the trivial solution
if and only if EVERY COLUMN of A is pivotal

Cannot have multiple vectors that have the same image

54
Q

Reflections through x1 axis, x2 axis, line x2 = x1, line x2 = -x1; the origin

A

1 0
0 -1
——-
-1 0
0 1
——-
0 1
1 0
———
0 -1
-1 0
———-
-1 0
0 -1

55
Q

Contractions and Expansions

A

if k is between 0 and 1: contraction
k > 1: expansion
Horizontal contraction & expansion:
k 0
0 1
————
Vertical contraction & expansion:
1 0
0 k

56
Q

Shears

A

Horizontal Shear:
1 k
0 1
Verticle Shear
1 0
k 1
if k is positive: right or up shear
if k is negative: left or down shear

57
Q

Projections

A

on the x1 axis:
1 0
0 0
on the x2 axis:
0 0
0 1

58
Q

Rotations COUNTERCLOCKWISE!!!!!!!!

A

cos(a) -sin(a)
sin (a) cos(a)

59
Q

Properties of Matrix Addition (6 total)

A

a. A + B = B + A d. r (A + B) = rA + rB
b. (A + B) + C = A + (B + C) e. (r + s) A = rA + sA
c. A + 0 = A f. r (sA) = (rs) A

60
Q

Matrix Multiplication

A

CAN ONLY MULTIPLE if the left matrix has the same number of columns as the number of rows of the RIGHT matrix

61
Q

Properties of Matrix Multiplication (5 total)

A

a. A (BC) = (AB) C
b. A (B + C) = AB + AC
c. (B + C) A = BA + CA
d. r (AB) = (rA) B = A (rB)
e. ImA = A = AIn

62
Q

matrices that commute

A

Matrices A and B commute if AB = BA

63
Q

Transposing Matrices

A

take the first row of the matrix and make it the first column
and continue like that

64
Q

WARNINGS when multiplying matrices

A

○ Order when multiplying matrices matters
■ In general, AB ≠ BA
○ AB = AC does not suggest B = C
○ If AB is the zero matrix, cannot conclude in general that either A = 0 or B = 0

65
Q

Properties of Matrix Transposing (4 total)

A

a. (AT)T = A
b. (A + B)T = AT + BT
c. For any scalar r, (rA)T = rAT
d. (AB)T = BTAT
○ The transpose of a product of matrices equals the product of their transposes in the reverse order

66
Q

Power of Matrices

A

only SQUARE matrices!!!!

67
Q

Uniqueness of RREF Theorem

A

each matrix is row equivalent to one and only RREF

68
Q

Theorem: Parametric Vector Form of a Nonhomogeneous System

A

● Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0.

69
Q

Theorem: Characterization of Linearly Dependent Sets

A

indexed set S= {v1…vp} of 2 or more vectors are linearly dependent if and only if at least one of the vectors in S is a linear combination of the others; if S is linearly dependent and v1 != 0, then some vj is a linear combination of the preceding vectors

70
Q

Theorem 8: Linear Dependence based on Matrix Size

A

More columns than rows means linear dependence

71
Q

Theorem : Using the Standard Matrix to find Columns of A

A

For T: Rn -> Rm (linear transformation), there exists a unique matrix A such that T(x) = ax for all x in Rn

T(ej), where ej is the jth column of the identity matrix in Rn
A = [T(e1) … T(en)]

72
Q

One-To-One use the Homogenous Equation Theorem

A

Let T: Rn -> Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution!!

Linear Algebra 1554 (3 decks)

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