Vocab

Question 1

linear equation

Answer 1

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

Question 2

system of linear equations/ linear system

Answer 2

collection of one or more linear equations involving the same variables

Question 3

linear systems are equivalent if

Answer 3

they have the same solution set

Question 4

two matrices are row equivalent

Answer 4

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

Question 5

leading entry

Answer 5

the leftmost nonzero number in a nonzero row

Question 6

basic variables

Answer 6

variables that correspond to pivot columns

Question 7

vector / column vector

Answer 7

matrix with only one column

Question 8

two vectors are equal if

Answer 8

their corresponding entries are equal

Question 9

linear combination

Answer 9

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

Question 10

homogeneous

Answer 10

Ax=0

Question 11

trivial solution

Answer 11

• when x=0

- every homogeneous equation has the trivial solution

Question 12

nontrivial solution

Answer 12

• x is nonzero

- occurs when there is a free variable

Question 13

linearly independent

Answer 13

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

Question 14

linearly dependent

Answer 14

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

Question 15

transformation / function / mapping

Answer 15

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

Question 16

Rn

Answer 16

domain

Question 17

Rm

Answer 17

codomain

Question 18

T(x)

Answer 18

image

Question 19

range

Answer 19

set of all T(x)

Question 20

a transformation is linear if

Answer 20

T(0) = 0
T(u+v) = T(u) + T(v)
T(cu) = cT(u)

Question 21

onto

Answer 21

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

Question 22

one-to-one

Answer 22

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

Question 23

two matrices are equal if

Answer 23

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

Question 24

nonsingular matrix

Answer 24

invertible matrix

Question 25

elementary matrices

Answer 25

obtained by performing one elementary row operation on the identity matrix

Question 26

subspace

Answer 26

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

Question 27

isomorphism from V onto W

Answer 27

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

Question 28

dim V

Answer 28

number of vectors in a basis for V

Question 29

Row A (row space)

Answer 29

set of all linear combinations of the row vectors

Question 30

rank

Answer 30

dimension of column space of A

Question 31

nullity

Answer 31

dimension of null space

Question 32

probability vector

Answer 32

vector with nonnegative entries that add up to 1

Question 33

stochastic matrix

Answer 33

square matrix whose columns are probability vectors

Question 34

steady-state vector / equilibrium vector

Answer 34

probability vector q such that Pq=q

Question 35

stochastic matrix is regular if

Answer 35

some matrix power P^k contains only positive entries

Question 36

eigenvector

Answer 36

NONZERO vector x such that Ax=(lambda)x

Question 37

eigenspace

Answer 37

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

Question 38

characteristic equation

Answer 38

det(A-(lambda)I)=0

Question 39

multiplicity

Answer 39

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

Question 40

two matrices A and B are similar if

Answer 40

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

Question 41

diagonalizable

Answer 41

matrix A is similar to a diagonal matrix

Question 42

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

Answer 42

repeller

Question 43

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

Answer 43

attractor

Question 44

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

Answer 44

saddle point

Question 45

for complex eigenvalues, if absolute value of eigenvalue > 1

Answer 45

spiral away from origin

Question 46

for complex eigenvalues, if absolute value of eigenvalue < 1

Answer 46

spiral toward origin

Question 47

``` if Re (lambda) > 0 ** differential equations ```

Answer 47

trajectories spiral outward

Question 48

if Re (lambda) = 0

Answer 48

trajectories form ellipses

Question 49

unit vector

Answer 49

vector whose length is 1

Question 50

two vectors are orthogonal if

Answer 50

their dot product is 0

Question 51

orthogonal complement

Answer 51

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

Question 52

orthonormal set

Answer 52

orthogonal set of unit vectors

Question 53

symmetric matrix

Answer 53

matrix such that the transpose of A = A

Question 54

orthogonal matrix

Answer 54

inverse of P = transpose of P; orthonormal columns

Question 55

spectrum

Answer 55

set of eigenvalues of a matrix

Question 56

positive definite

Answer 56

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

Question 57

negative definite

Answer 57

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

Question 58

indefinite

Answer 58

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

Question 59

positive semidefinite

Answer 59

Q(x) >= 0 for all x

Question 60

negative semidefinite

Answer 60

Q(x) <= 0 for all x

Question 61

singular values of A

Answer 61

square roots of the eigenvalues of A