Vocab Flashcards

1
Q

linear equation

A

an equation that can be written in the form a1x1 + … + anxn = b with b and the coefficients a1…an are real/complex numbers

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2
Q

system of linear equations/ linear system

A

collection of one or more linear equations involving the same variables

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3
Q

linear systems are equivalent if

A

they have the same solution set

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4
Q

two matrices are row equivalent

A

if there is a sequence of elementary row operations that transforms one matrix to another

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5
Q

leading entry

A

the leftmost nonzero number in a nonzero row

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6
Q

basic variables

A

variables that correspond to pivot columns

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7
Q

vector / column vector

A

matrix with only one column

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8
Q

two vectors are equal if

A

their corresponding entries are equal

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9
Q

linear combination

A

y = c1v1 + … + cnvn given vectors v1,…,vn, and scalars c1,…,cn

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10
Q

homogeneous

A

Ax=0

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11
Q

trivial solution

A
  • when x=0

- every homogeneous equation has the trivial solution

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12
Q

nontrivial solution

A
  • x is nonzero

- occurs when there is a free variable

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13
Q

linearly independent

A

set of vectors {v1, … , vp} with the vector equation x1v1+ … + xpvp = 0 with only the trivial solution

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14
Q

linearly dependent

A

there exists weights c1, …, cp not all zeros such that c1v1 + …+ cpvp = 0

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15
Q

transformation / function / mapping

A

a rule that assigns each vector x in Rn a vector T(x) in Rm

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16
Q

Rn

A

domain

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17
Q

Rm

A

codomain

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18
Q

T(x)

A

image

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19
Q

range

A

set of all T(x)

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20
Q

a transformation is linear if

A
T(0) = 0
T(u+v) = T(u) + T(v)
T(cu) = cT(u)
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21
Q

onto

A

each b in Rm is the image of at least one x in Rn

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22
Q

one-to-one

A

each b in Rm is the image of at most one x in Rn

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23
Q

two matrices are equal if

A

they have the same size (same number of rows and columns)

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24
Q

nonsingular matrix

A

invertible matrix

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25
Q

elementary matrices

A

obtained by performing one elementary row operation on the identity matrix

26
Q

subspace

A
  1. includes the zero vector
  2. is closed under vector addition
  3. is closed under scalar multiplication
27
Q

isomorphism from V onto W

A

one-to-one transformation from a vector space V onto vector space W

28
Q

dim V

A

number of vectors in a basis for V

29
Q

Row A (row space)

A

set of all linear combinations of the row vectors

30
Q

rank

A

dimension of column space of A

31
Q

nullity

A

dimension of null space

32
Q

probability vector

A

vector with nonnegative entries that add up to 1

33
Q

stochastic matrix

A

square matrix whose columns are probability vectors

34
Q

steady-state vector / equilibrium vector

A

probability vector q such that Pq=q

35
Q

stochastic matrix is regular if

A

some matrix power P^k contains only positive entries

36
Q

eigenvector

A

NONZERO vector x such that Ax=(lambda)x

37
Q

eigenspace

A

set of all solutions of (A-(lambda)I)x=0

38
Q

characteristic equation

A

det(A-(lambda)I)=0

39
Q

multiplicity

A

number of times it appears as a root of the characteristic equation

40
Q

two matrices A and B are similar if

A

there is an invertible matrix such that A=PBP^-1

41
Q

diagonalizable

A

matrix A is similar to a diagonal matrix

42
Q

if all eigenvalues are greater than 1, the origin is a

A

repeller

43
Q

if all eigenvalues are less than 1, the origin is a

A

attractor

44
Q

if some eigenvalues >1 and others <1, the origin is a

A

saddle point

45
Q

for complex eigenvalues, if absolute value of eigenvalue > 1

A

spiral away from origin

46
Q

for complex eigenvalues, if absolute value of eigenvalue < 1

A

spiral toward origin

47
Q
if Re (lambda) > 0
** differential equations
A

trajectories spiral outward

48
Q

if Re (lambda) = 0

A

trajectories form ellipses

49
Q

unit vector

A

vector whose length is 1

50
Q

two vectors are orthogonal if

A

their dot product is 0

51
Q

orthogonal complement

A

set of all vectors z that are orthogonal to every vector in subspace W

52
Q

orthonormal set

A

orthogonal set of unit vectors

53
Q

symmetric matrix

A

matrix such that the transpose of A = A

54
Q

orthogonal matrix

A

inverse of P = transpose of P; orthonormal columns

55
Q

spectrum

A

set of eigenvalues of a matrix

56
Q

positive definite

A

Q(x) > 0 for all x =/= 0

57
Q

negative definite

A

Q(x) < 0 for all x =/= 0

58
Q

indefinite

A

Q(x) <0 and >0 for all x =/= 0

59
Q

positive semidefinite

A

Q(x) >= 0 for all x

60
Q

negative semidefinite

A

Q(x) <= 0 for all x

61
Q

singular values of A

A

square roots of the eigenvalues of A