Matrix Test 2 2.2-2.3, 2.5, 3.1-3.3

When A is invertible, the detA =

(-1)^r * (product of diagonal entries in echelon matrix "U") where r is the number of interchanges from A to U

(A^T)^-1 =

(A^-1)^T

det(A^n) =

(det(A))^n

zero subspace

-the set consisting of only the zero vector in a vector space V is a subspace of V written as: {0}

For each u in V, and scalar c, 0*u = c*0 = (-1)*u =

0;0;-u

what is the determinant of an elementary row replacement matrix? (with a k below the diagonal in identity matrix)

1

If A is invertibe, then detA^-1 =

1 / detA

LU Factorization Algorithm

1) Row reduce A to echelon form (That is your U value) 2)Set up L such that there are ones down the diagonal and zeroes above. 3)Column one of L is equal to column one of A divided by its pivot position 4) Column two of L is equal to the column two of A once the first column has zeroes beneath the pivot, then divided by its top entry. 5) the other columns are formed the same way, resulting in L

The following axioms must hold for all vectors, u, v, and w in V and for all scalars c and d

1) The sum of u+v is in V 2) u+v = v+u 3) (u+v)+w = u+(v+w) 4)There is a zero vector 0 in V such that u+0=u (unique to that vector) 5)For each u in V, there is a vector -u, in V such that u+(-u) = 0 (unique to that vector) 6)The scalar multiple of u by c, denoted cu is in V 7)c(u+v) = cu+cu 8)(c+d)u=cu+du 9)c(du)=(cd)u 10)1u=u

Properties of a subspace that need to be checked; if they're true, the other seven are automatically satisfied. A subspace of a vector space V is a subset H of V that has 3 properties: What do these properties prove about H?

1) The zero vector of V is in H^2 2) H is closed under vector addition (u+v is in H) 3) H is closed under mutliplication by scalars (cu is in H) **These properties guarantee that a subspace H of V is itself a vector space

Formula for INVERSE of a 2x2 matrix

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

LU =

A

when A^T is singular, so is

A

Let A and B be square matrices. If AB = I, then

A and B are both invertible with B=A^-1 and A=B^-1

An nxn matrix A is invertible iff

A is row equivalent to In

The column space of an mxn matrix A is all of R^m iff

Ax=b has a solution for each b in R^m

the (i, j) cofactor equation and determinant equation with cofactor in it

Cij = (-1)^i+j detAij detA = a11C11+a12C12+...+a1nC1n

How to find A^-1 with LU factorization

Determine LU, find the inverse of U and the inverse of L, then multiply U^-1 * L^-1 and thats equal to A^-1

If an elementary row operation is performed on an mxm matrix A, the resulting matrix can be written as

EA

A square matrix needs to be in ___________ form in order to find the determinant by multiplying the diagonal terms.

Echelon, Not RRef

Finding the column space

Find rref, write in parametric vector form, column space is the columns of the free variables of the original A

A^-1 X A = A X A^-1 =

I

each elementary matrix is invertible, so the inverse of E is of the same type that transforms E back into

I

If A is an invertible matrix. then A^T is _______________.

Invertible

LU Factorization Equations

Ly = b Ux = y

Is the echelon form of U unique? Are the pivots unique? Is the product of pivots unique?

No; no; yes

The conclusions in Theorem 10 hold whenever S is in a region in ____ with finite area or a region in ____ with finite volume

R^2, R^3

The Algorithm for finding A^-1

Row reduce the augmented matrix [A I]. If A is row equivalent to I, then [A I] is row equivalent to [I A^(-1)]. Otherwise, A does not have an inverse.

A linear transformation T: Rn to Rn is said to be invertible if there exists a function S: Rn to Rn such that: S is considered the ______________ of T. Therefore, S is _____________ and a ___________________ _________________

S(T(x)) = x for all x in rn T(S(x)) = x for all x in rn inverse unique; linear transformation

If v1....vp are in a vector space V, then

Span{v1...vp} is a subspace of V

Facts about the explicit NulA

The spanning set is linearly independent since theyre written in terms of the free variable. When NulA contains nonzero vectors, the number of vectors in the spanning set for NulA equals the number of free variables in the equation Ax=0.

L^-1 * A =

U

To find the area of a 2x2 parallelogram, set up the matrix as: ____________ and the area is

[ x1 x2] [y1 y2] of two verticies that represent the vectors, not the vector u+v area = det[ a 0] [ 0 d]

Determinabnt of a 1x1 matrix

[a11]

elementary matrix

a matrix obtained by performing a single elementary row operation on an identity matrix

nonsingular matrix

a matrix that is invertible

singular matrix

a matrix that is not invertible

vector space

a nonempty set V, of objects called vectors, on which are defined two, called addition and multiplication by scalars

The column space of an mxn matrix A is

a subspace of R^m

The invertible matrix theorem: Let A be a square nxn matrix. Then the following statements are equivalent. That is, the statements are either all true or all false (12)

a) A is an invertible matrix b) A is row equivalent to the nxn identity matrix c) A has n pivot positions d)Ax=0 has only the trivial solution e)The columns of A form a linearly independent set f)The linear transformation x to Ax is one-to-one g) Ax=b has at least one solution for each b in rn h)The columns of A span Rn i)The LT x to Ax maps rn onto rn j) There is an nxn matrix C such that CA=I k) There is an nxn matrix D such that AD=I l)A^T is an invertible matrix m) detA does not equal 0

determinant of a 3x3 matrix

a11*detA11 - a12*detA12 + a13*detA13 where A11 is obtained from A by deleting the first row and the first column and finding the determinant of that 2x2 matrix

Determinant of a 2x2 matrix formula

ad-bc

Let T: Rn to Rn be a linear transformation and let A be the standard matrix for T. Then T is invertible iff A is ____ ___________ ___________. In that case, the linear transformation S is given by S(x) =

an invertible matrix ; A^-1x (unique)

determinant equation

det A = a11detA11 - a12DetA12 + ... + (-1)^1+n a1ndetA1n

A square matrix A is invertible if and only if

det(A) does not equal 0

If A is an nxn matrix, detA^T =

detA

If A and B are nxn matricies, detAB =

detA * detb

Det(A+B) does NOT equal

detA + detB

cofactor expansion across jth column

detA = a1jC1j + a2jC2j + ... +anjCnj **equals a number not a matrix**

Cofactor expansion across first row equation

detA = ai1 Ci1 + ai2 Ci2 + ...+ ainCin **** equals a number!!!!! not a matrix

If two rows (or columns) of a square matrix are interchanged to produce B, then how are the determinants of A and B related?

detB = -detA

If a multiple of one row (or column) of A is added to another row (or column) to produce B then, how is the determinant of A and B related?

detB = detA

If one row (or column) of a square matrix A is multiplied by k to produce B, how are their determiants related?

detB = kdetA

detA = (-1)^r times

detU (echelon form of A) (where r is the number of row interchanges to get to echelon form)

Finding the nul space

do rref of the matrix augmented with 0 write in parametric vector form

The factorization of a matrix A:

expresses A as a product of two or more matricies.

In order for a plane or line to be a subspace of R^3 or R^2, it must: Why?

go through the origin because if the plane or line didnt, the zero vector wouldn't be a part of H

A linear transformation T from a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W such that:

i) T(u+v)=T(u)+T(v) for all u, v in V ii) T(cu) for all u in V and all scalars c

Parallelogram rule for addition

if u and v are in R^2, and represented as points in the plane, then u+v is the fourth vertex whose other verticies are 0, u, and v

If A and B are invertible matricies, then AB is ______________.

invertible

since row operations are reversible, elementary matricies are ___________

invertible

If A is an invertible matrix, then A^-1 is ___________ and (A^-1)^-1=

invertible; A

If a matrix is invertible, then

it is an upper or lower triangular matrix with ones on the diagonal

For a 2x2 matrix, it is invertible if

its determinant does not equal zero

What is the determinant of an elementary scaling matrix with k on the diagonal? (k in diagonal of identity matrix)

k

If a matrix has a zero row or column, it is

linearly dependent

The determinant of A equals 0 when its rows and columns are ____________________ _____________________.

linearly dependent

when p is a scalar, det(pA) =

p^n*det(A)

Determinant of a triangular matrix

product of diagonal entries

ColA is the __________ of the LT of x onto Ax

range

Let S be the space of all doubly infinite sequences of numbers: {yk}=(...y-2, y-1, yo, y+1, y+2). If {zk} is another element of S, then the sum {yk} + {zk} is the _____________, {yk +zk}. the scalar multiple c{yk} is the _______________, {cyk}. S is called the

sequence; sequence; space of signals

linearly dependent columns of A^T make A^T

singular

Finding the NulA (explicit definition)

solve the matrix for Ax=0 and write in parametric vector form

The invertible matrix theorem only applies to _____________ matricies

square

Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a ____________ of R^n.

subspace

The null space of an mxm matrix A is a __________________ of R^n.

subspace

every vector space is a

subspace

When a1 and a2 are nonzero vectors, for any scalar c, the area of the parallelogram determined by a1 and a2 is equal to

the area of the parallelogram determined by a1 and a2+ca1

The volume of a 3x3 mtrix is zero when

the columns are linearly dependent

If ad-bc =0, then

the matrix is not invertible

The inverse of a product is equal to

the product of their inverses in reverse order (AB)^-1 = B^-1 A^-1

The column space

the set of all linear combinations of the columns of A (span of the columns of A)

The null space

the set of all solutions of the homogeneous equation Ax=0

Nul A in terms of a transformation

the set of all x in R^n that are mapped into the zero vector of R^m

subspace

the subset of vectors from a larger vector space

An nxn matrix is said to be invertible if

there exists an nxn matrix C such that AC=I AND CA=I

True or false: the column space of an mxn matrix is in R^M

true

The term subspace is used when at least __________ ______________ __________________ are in mind, with ____________ inside the ______________.

two vector spaces; one; other

If a matrix has an inverse, the inverse is considered to be

unique

every sub space is a

vector space

The null space is a ________ ____________

vector space/ subspace

A^-1*Ax =

x

If A is an invertible nxn matrix, then for each b in R^n, the equation Ax=b has the unique solution:

x = A^-1 * b

ColA=

{b:b = Ax for some x in R^n} where Ax is a LC of the columns of A

NulA in set notation

{x:x is in R^n and Ax=0}

If A is a 3x3 matrix, the area of the volume of the parallelpiped (figure between two planes) determined by the columns of A is

|detA|

If A is a 2x2 matrix, the area of the parallelogram determined by the columns of A is

|detA| (in echelon form)

Theorem 10: Let T: R^2 onto R^2 be the linear transformation determined by a 2x2 matrix. If S is a parallelogram in R^2, then {area of T(S)} =

|detA|*{area of S}

Also Theorem 10: If T is determined by a 3x3 matrix A, and if S is a parallelepiped in R^3, then {Volume of T(S)} =

|detA|*{volume of S}