test 2 vocab

Question 1

closure property for vector addition

Answer 1

x+y in V for all x,y in V

Question 2

closure property for scalar multiplication

Answer 2

ax in V for all a in F and x in V

Question 3

subspace

Answer 3

Let S be an nonempty subset of a veto space V over F. If S is a vector space over F using the addition and scalar multiplication operations, then S is said to be a subspace of V.

Question 4

trivial subspace

Answer 4

given a vector space V, the set Z={0} containing only the zero vector is a subspace of V because (A1) and (M1) are trivially satisfied.

Question 5

parallelogram law

Answer 5

Vector addition in R2 and R3 is easily visualized by using the parallelogram law, which states that two vectors U and V, the sum U+V is the vector defined by the diagonal of the parallelogram.

Question 6

space spanned by S

Answer 6

for a set of vectors S ={v1,v2…vr}, the subspace span(S)= {a1v1+a2v2+…arvr} generated by forming all linear combinations of vectors from S.
If V is a vector space such that V=span(S), we say S is a spanning set for V. in other words, S spans V whenever each vector in V is a linear combination of scots from S

Question 7

Sum of subspaces

Answer 7

if X and Y are subspaces of a vector space V, then the sum of X and Y is defined to be the set of all possible sums of vectors from X with vectors from Y. The sum X+Y is again a subspace of V. If Sx, Sy span X,Y then Sx U Sy spans X+Y

Question 8

Range

Answer 8

For a linear function f mapping Rn into Rm, let R(f) denote the range of f. That is, R(f)= {f(x)|x is in Rn} is a subset of Rm is the set of all “images” as x varies freely over Rn

Question 9

linear spaces

Answer 9

subspaces of Rm

Question 10

range of a matrix

Answer 10

A is in Rmxn is defined to be the subspace R(A) of Rm that is generated by the range f(x)=Ax

Question 11

image space of A

Answer 11

Because R(A) is the set of all “images” of vectors x is in Rm under transformation by A, some people call R(A) the image space of A

Question 12

nullspace

Answer 12

N(f)- the set of vectors that are mapped to 0.

Question 13

nullspace of A

Answer 13

N(A)- the set of all solutions to the homogeneous system Ax=0

Question 14

left hand nullspace

Answer 14

N(A^T)- the set of all solutions to the left hand homogenous system y^TA=0^T

Question 15

linearly independent set

Answer 15

A set of vectors is said to be linearly independent set whenever the only solution for the scalars ai in the homogeneous equation: a1v1+a2v2+…+anvn=0 is the trivial solution a1=a2=…an=0. Those that contain no dependency relationships. empty set is always linearly independent

Question 16

linearly dependent set

Answer 16

Whenever there is a nontrivial solution for the a’s (at least one ai is not 0) in the set S. Those in which at least one vector is a combination of the others

Question 17

diagonally dominant

Answer 17

The magnitude of each diagonal entry exceeds the sum of the magnitude of the off diagonal entire in the corresponding row.

Question 18

maximal linearly independent subset of columns

Answer 18

a linearly independent set containing as many columns from A as possible. The basic columns in A always constitute one solution

Question 19

basis

Answer 19

A linearly independent spanning set for a vector spanning set for a vector space V

Question 20

least squares solution

Answer 20

any vector that provides a minimum value for (Ax-b)^T(Ax-b)

Question 21

euclidean norm

Answer 21

||x|| = (x^Tx)^(1/2) for real and ||x|| = (x*x)^(1/2) for complex

Question 22

normalize

Answer 22

we normalize x by setting u=x/||x||

Question 23

distance

Answer 23

can be visualized with the aid of the parallelogram law||u-v||

Question 24

triangle inequality

Answer 24

||x+y|| is less than or equal to ||x|| + ||y||

Question 25

unit space spheres

Answer 25

S1 is a an octahedron , S2 is a sphere and S infinity is a cube

Question 26

norm

Answer 26

for a real or complex vector space V is a function ||*|| mapping V into R that satisfies
||x||>=0 ||x||=0 x=0
||ax||=|a| ||x|| for all scalars a
||x+y|| <= ||x||+||y||

Question 27

frobenius matrix norm

Answer 27

||A||^2 F= trace(A*A)

Question 28

matrix norm

Answer 28

a function ||*|| from the set of all complex matrices into R that satisfies… similar properties to the nor, and also ||AB||<= ||A|| ||B|| for all compatible matrices

Question 29

induces

Answer 29

A vector norm that is defined on Cp for all p=m,n induces a matrix norm on Cmxn by setting ||A||=max(||x||=1) ||Ax||

Question 30

A- inner product

Answer 30

If Anxn is a nonsingular matrix then =xAAy is an inner product for Cnx1

Question 31

standard inner product for matrices

Answer 31

<a>= trace(A^TB) and </a><a>=trace(A*B)</a>

Question 32

A-nrom

Answer 32

generated by the A inner product on Cnx1 is ||x||A= (A|A)^(1/2) = trace(A*A)^(1/2) =||A||f

Question 33

parallelogram identity

Answer 33

||x+y||^2 +||x-y||^2 = 2(||x||^2 +||y||^2)

Question 34

orthogonal

Answer 34

in an inner product space V, 2 vectors are orthogonal whenever=0. for real numbers, x^Ty=0 and for complex numbers, x*y=0

Question 35

angle

Answer 35

in a real inner product space the radian measure of the angle between nonzero vectors is defined to be cos()=/(||x|| ||y||)

Question 36

orthonormal set

Answer 36

whenever ||ui|| = 1 for each i and ui is perpendicular to uj for all i not equal to j. every orthogonal set is linearly independent.

Question 37

Fourier expansions

Answer 37

if B={u1,u2….un} is an orthonormal basis for an inner product space V, then each x in V can be expressed as
x=u1 + u2+…un. The scalars are the coordinates of x with respect to B and they are called Fourier coefficients.

Question 38

Gram-Schmidt sequence

Answer 38

if B={x1’x2….xn} is a basis for a general inner product space S tenth GS sequence is defined by:
u1=x1/||x1|| and uk=…. (see notes) is an orthonormal basis for S.

Question 39

QR factorization

Answer 39

factorization A=QR, it is uniquely determined by A. Every matrix Anxm with linearly independent columns can be uniquely factored as A=QR in which the columns of Qmxn are an orthonormal basis for R(A) and Rnxn us an upper triangular matrix with positive diagonal entires. It is the complete road map of the GS process because the columns of Q=(q1|q2|….|qn) are the result of applying the GS procedure to the columns of A=(a1|a2|…|an) and R is given by… (see notes)

Question 40

permutation

Answer 40

A permutation p = (p1,p2,…,pn) of the numbers (1,2,…,n) is simply any rearrangement.

Question 41

Parity

Answer 41

The parity of a permutation is unique—i.e., if a permuta- tion p can be restored to natural order by an even (odd) number of interchanges, then every other sequence of interchanges that restores p to natural order must also be even (odd).

Question 42

sign of a permutation

Answer 42

+1: if p can be restored to natural order by an even number of interchanges,
-1: if p can be restored to natural order by an odd number of interchanges.

Question 43

minor determinant

Answer 43

of Am×n is defined to be the determinant of any k × k submatrix of A

Question 44

Schur complements

Answer 44

The matrices D − CA^(−1B) and A − BD^(−1C)

Question 45

cofactor

Answer 45

of An×n associated with the (i, j)-position is defined as ̊Aij=(−1)^(i+)jMij,
where Mij is the n−1 × n−1 minor obtained by deleting the ith row and jth column of A. The matrix of cofactors is denoted by ̊A.

Question 46

adjugate

Answer 46

of An×n is defined to be adj (A) = ̊A^T , the transpose of the matrix of cofactors. If A is nonsingular, then
A^−1 = ( ̊A^T)/det(A)= adj(A)/det(A).

Question 47

unitary matrix

Answer 47

a complex matrix U(nxn) whose columns (or rows) constitute an orthonormal basis for C^n.

Question 48

orthogonal matrix

Answer 48

a real matrix P(nxn) whose columns(or rows) constitute an orthonormal basis for R^n.