Concepts and definitions

Question 1

f: x –> y is invertible if and only if

Answer 1

1. for every y in Y there exists a unique x in X such that f(x) = y (this is bijection)

Question 2

Subspace

Answer 2

1. Contains the zero vector
2. closure under scalar multiplication
3. closure under addition

Question 3

Definition of invertible

Answer 3

for f: R^n to R^m the function g: R^m to r^n is the inverse of f if
1. f * g(x) = x
2. g * f(x) = x

Question 4

How to check invertible?

Answer 4

ad-bc != 0

Question 5

how to calculate invertible

Answer 5

1/ad-bc * [d -b] [-c a]

Question 6

Transpose properties

Answer 6

1. (AT)*T = A
2. (A+B)T = AT+BT
3. (rA)T = r* AT
4. A(BT) = BT* AT

Question 7

Surjective (onto)

Answer 7

for every y in Y there exists at least one x in X such that f(x) = y or the image

Question 8

Injective (one to one)

Answer 8

for any y in Y there exists at most one x in X such that f(x) = y

Question 9

Bijection

Answer 9

for every y in Y there exists a unique x in X such that f(x) = y

Question 10

Span

Answer 10

set of all possible linear combinations of the vectors

Question 11

Basis of subspace

Answer 11

the minimum set of vectors that spans the subspace
1. spans R^n
2. linearly independent

Question 12

nullspace of A

Answer 12

set of vectors x in R^n (column vectors) such that A * set equals the zero vector

Question 13

column space of A

Answer 13

1. set of all linear combinations of column vectors in A
2. subspace of R^m (dependent on pivot columns, less than or equal to codomain)
3. pivot columns of A form basis

Question 14

dimension

Answer 14

of column vectors in any basis of H

Question 15

rank

Answer 15

dimension of column space of A or number of pivot columns

Question 16

Rank nullity theorem

Answer 16

if A^t has n columns, that rank(A) + dim(nul(A)) = n

Question 17

Determinants

Answer 17

quantity determined by ad-bc, defined for matrices nxn, measure of distortion

Question 18

Properties of Determinants

Answer 18

1. det(A*B) equals det(A) * det(B)
2. in A, row is scaled by k, det(B) = k* det(A)
3. in A, two rows are swapped, det(B) = -det(A)
4. in A, row is replaced by combination of another row, det(B) = det(A)

Question 19

Area or volume of a parallelogram

Answer 19

find determinant

Question 20

eigenvector v

Answer 20

if for some scalar lambda exists such that Av = lambdav

Question 21

eigenvalue

Answer 21

exists if and only if:
1. some vector x exists such that Ax = lambdax
2. Av - lambdav = zero vector
3. Ax - (lambdaI)*x = zero vector

Question 22

Eigenspace

Answer 22

Since we find the nullspace associated with the matrix (A-lambda*I) then we have found a subspace of A associated to lambda

Question 23

length of a vector

Answer 23

sqrt(v1^2+ …. + vn^2)

Question 24

distance from u to v

Answer 24

sqrt( (u1 - v1)^2 + ……. + (un - vn)^2)

Question 25

dot product of u and v (vectors)

Answer 25

u^T * v

Question 26

orthogonal (perpendicular)

Answer 26

when two vectors multiplied by dot product = 0

Question 27

vector projection

Answer 27

proj of y onto u = (y*u)/length of u

Question 28

Gram-Schmidt theorem (orthogonal basis)

Answer 28

a basis: span(v1 to vn) for subspace V such that i !=j, vi * vj = 0

Question 29

triangular matrix

Answer 29

elements above or below diagonal are zero

Question 30

diagonal matrix

Answer 30

elements above and below diagonal are zero

Question 31

Image(f)

Answer 31

subset you do map to (surjective)

Question 32

consistent or non-consistent

Answer 32

if RREF(A) has either 1 or many solutions, can be reduced down to a triangular form

Question 33

finding an eigenvalue

Answer 33

det(A-lambda*I) = 0

Question 34

checking an eigenvector

Answer 34

Av = lambdav, set vector with matrix and solve, if scalar multiple then yes

Question 35

finding eigenvectors

Answer 35

(A-lambaI)x = 0, set adjust matrix to 0 and solve, find free variable vectors

Question 36

finding a basis

Answer 36

(A-lambaI)x = 0, same thing as finding eigenvectors