Vectors Test 2

(AB^T)-1

(B^-1)T x A-1

what if there is a row of all zeros on the bottom (independent or dependent)

- can still be independent -cannot be generating set

If A is the identity matrix with one row switched, what is the determinant?

-1

what do you do to show if something is not a linear transformation

-choose random numbers

put into different words "suppose we want to determine if w is in the range of T"

-equivalent to asking if there exists a vector x such that T(x) = w

Theorem 2.11 (definition one-to-one)

-every pair of distinct vectors in R^n has distinct images in R^m 1. T is one-to-one 2. The null space of T consists only of the zero vector 3. the columns of A are linearly independent 4. rank A = n (the number of columns) 5. pivot position in each column (can have all zeros on bottom) -every pair of distinct vectors in R^n has distinct images in R^m

Theorem 2.10 (onto) when is a function said to be onto? how would you solve this? What stands out as you are reducing to see if it is one-to-one

-if its range is all of R^m; every vector in R^m is an image -the span of the columns is R^m (the columns are a generating set) -pivot position in each row -rref has no zero rows -rank of A=m (# of rows) -Ax=b has at least one solution rref form and see if the rank is equal to the rows -rref has no zero rows

definition of an elementary matrix

-if you can obtain the elementary matrix by a single elementary row operation of the identity matrix

what does it mean if you find a row of all zeros on the bottom?

-then there is some vector b that could make the set inconsistent so it cannot be a generating set

Identity Transformation Codomain and Domain

-these will be the same

Calculating the area of a parallelogram determined by u and v when given two vectors

-this is determined by two vectors in R2 -find the absolute value of the determinant of the two vectors combined -must be R2, R3 is the volume of a parallelepiped

Theorem 2.6 (9 long matrix invertibility statements)

1. A is invertible 2. rref = In 3. rank equals number of columns 4. span of the columns is the number of columns is (R^n) 5. equation is consistent for every b 6 nullity= 0 (n-rank) 7. linearly independent 8. Ax=0 is 0 9. A is the product of an n x n elementary matrix

Theorem 3.4 (invertibility of a matrix: if and only if, det AB, det A^T, DetA^-1)

1. A is invertible if an only if det A ≠ 0 2. detAB = (detA)(detB) 3. det A^T=detA (determinant of a transpose is the same as detA) 4. If A is invertible, then detA-1 = 1/detA (the determinant of the inverse is 1/detA)

Theorem 2.2 (3 properties/formulas of matrix inverses) Inverse of A^-1 inverse of AB^-1 inverse of Atranspose

1. If A is invertible then A^-1 is invertible and (A^-1)^-1=A -the inverse of the inverse of A is A 2. (AB)^-1 = B^-1A-1 if A and B are both invertible 3. If A is invertible, then A transpose is invertible -the inverse of A transpose is A inverse transposed

Theorem 2.13 Linear Transformation Inverses

1. T is a linear transformation with standard matrix A 2. T is invertible if and only if A is invertible 3. T^-1=Ta-1 4. T inverse is linear 5. It's standard matrix is A-1

5 properties of linear dependent and independent sets

1. a set consisting of a non zero vector is dependent 2. a set of two vectors is dependent if and only if they are multiples 3. more columns than rows implies dependence 4. if no vector can be removed without changing its span, its linear independent

theorem 2.4 (column correspondence)

1. pivot columns of A are linearly independent 2. each non pivot column is a linear combination of the previous pivot columns of A, where the coefficients are the single entry of the corresponding column of rref(A)

theorem 2.7 (two truths for any mxn matrix A)

1.Ta (u+v) = Ta(u) +Ta(v) 2.Ta(cu) = cTa(u)

What is the size of the standard matrix R3 to R2

2 x 3

what are you saying if you say S is a generating set...what shows you that something is not a spanning set?

Ax=b has a solution (is consistent) for all b where A is the matrix whose columns are vectors in S - a row of all zeros

relating all three topics (linear transformations, matrices, and systems of linear equations)

Ax=b has a solution if and only if b is in the range of Ta Ax=b has a solution for every b if and only if Ta is onto Ax=b has at most one solution for every b if and only if Ta is one-to-one

two linear transformations that deserve special attention

Identity Transformation -I is linear and its range is all of R^n I(x)=x -identity transformation is onto Zero transformation -T0(x)=0 -linear -range consists precisely of the zero vector

Theorem 3.3 (rules of determinant operations)

Let A be an n x n (Square Matrix) 1. interchange two rows of A, then DetB= -det A (negate it) 2. multiplying by a scalar, then DetB= kdetA 3. adding a multiple of one row to another does nothing 4. an n x n elementary matrix E, detEA= (detE)(detA)

definition of a standard matrix (Theorem 2.9)

Let T: Rn to Rm be linear. Then there is a unique m x n matrix A= [T(e1) T(e2).. T(en)] whose columns are the images under T of the standard vectors such that T(v)= Av

Define Codomain (in terms of images)

R^m number of rows in the matrix (Rn to Rm) -the set of all images f(x)

Domain

R^n number of columns in the matrix (Rn to Rm) -the first function

Find a matrix such that AB=0 but BA ≠ 0

Set B= 1 1 Set A= 2 -2 1 1 1 -1

How do you show if T is invertible? How do you determine T-1

T is invertible if and only if its standard M

what does it mean if the zero vector is the only vector whose image under T is the zero vector

T is linear independent and T must be one to one

chart on 188 (number of solutions, property of the columns, property of the rank

T is onto Ax=b has at least one solution (Ax=b is consistent) The columns of A are a generating set for Rm rankA=m (the number of rows ) T is one-to-one Ax=b has at most one solution for every b in Rm (no free variables) the columns of A are linearly independent rankA=n (the number of rows) T is invertible Ax=b has a unique solution for every b in Rm columns are a linearly independent generating set for Rm

Theorem 2.8 (4 true statements for linear transformations)

T(0)=0 (zero transformation = 0) T(-u)= -T(u) T(u-v)= T(u)-T(v) T(au+bv) = aT(u) =bT(v)

Theorem 2.12 (composition rule)

T:Rn to Rm (matrix A) U: Rm to Rp (matrix B) UT Rn to Rp has standard matrix BA

the image of the zero vector under any linear transformation is the zero vector (true false)

This is true by the four rules of linear transformations in theorem 2.8

If T and U are linear transformations whose standard matrices are equal then T and U are equal

True. Standard matrices are unique

If T(u+V) = T(u) + T(v) for all vectors u and v in the domain of T then T is said to preserve vector addition

True. This is the definition

linear transformation

a linear transformation is a function that preserves the vector addition rule and scalar multiplication rule for all vectors in R^n and all scalars c 1. preserves vector addition T(u+v)= T(u)+T(v) -sum of two vectors is the sum of their images 2. preserves scalar multiplication T(cu)= cT(u)

define a function f from S1 to S2

a rule that assigns each vector in S1 and a unique vector f(v) in S2 the vector it assigns is called the image S1 is the domain S2 is the codomain range is the set of images for all vectors in S1

give an example of an invertible matrix

all elementary matrices are invertible and so is the identity matrix

when does Ax=b have a solution (if and only if)

b is in the range of the Ta

the zero vector (independent or dependent)

dependent

what if there is a free variable (independent or dependent)

dependent

What is the area of the parallelogram that under goes a transformation

det A x det[u v] -where A is the standard matrix of T -the area of the parallelogram is detA times larger

every matrix transformation is linear

every matrix transformation is linear because they follow the distributive and associative properties (vector addition and scalar mult)

If A and B are invertible matrices then A+ B is invertible

false

If A and B are invertible n x n matrices then AB is also invertible (true or false)

false

Every function preserves scalar multiplication (true false)

false.

the domain of a function is the set of all images f(x) (true false)

false. The set of all images is the codomain

The set [T(e1), T(e2)...] is a generating set for the range of any function (true false)

false. not necessarily true for non-linear functions

the codomain of any function is contained in its range (true false)

false. the range is contained in the codomain

if f is a function and f(u)=f(v), then u= v (true false)

false. the vectors are not necessarily equal. Two distinct vectors can have the same image

when does Ax=b have at most one solution for every b

if and only if Ta is one-to-one

how is a set of two vectors linearly dependent

if and only if one of the vectors is a multiple of the other

when is matrix A a generating set? 2nd way to notice this?

if the rank is equal to the number of rows -with two vectors, non-parallel works too

what's an example where not every vector r2 is an image of a vector in R3...how do you prove something is not a linear transformation?

if the rule provides an x^2 so the images has to be nonnegative. To prove this, use random vectors 'x1+x2+x3 x1^2

use of T(0)= 0 in terms of whether or not a function is linear

if this condition is not satisfied then the function is not linear -if the function is satisfied it does not necessarily tells us whether or not it is linear

a set consisting of a single nonzero vector (independent or dependent)

independent

What does it say about solutions if t is not onto (rank does not equal number of rows)

it is not consistent for every b (there exists a vector b that is not in the range)

if a square matrix has a column or a row consisting of all zeros, then...

it is not invertible

what does having more columns than rows imply? (independent or dependent)

linear dependence

a set that contains the zero vector (independent or dependent)

linear dependent!

when does Ax=b have a solution for every b (if and only if)

only if Ta is onto

R2 vs R3

plane and space

how to simply make a transformation that is not linear

put a constant in it

what is the range of a linear transformation

span of the columns of its standard matrix

In order for a composition UT of functions to be defined...

the codomain of the second function T should be the domain of the first function U T: Rn to Rm U: Rp to Rq

What does it say if Ax=b is consistent for every b in Rn

the rank is A, and further more Ax=b has a unique solution for every b

what is the range of a function

the set of images f(v) for all v in S1

define nullspace

the set of vectors in Rn whose image is 0

what does it says about solutions if t is not one-to-one

the solution is never unique

define what the generating set in the null space of T is

the vectors in the general solution

If AB=BA=In and AC=CA=In

then B=C because there is a unique solution

a function that preserves scalar multiplication is linear (true false)

this is false because it also needs to preserve vector addition

every function has a standard matrix (true false)

this is false because the function is not necessarily defined for the standard vectors (functions with zeros and exponents)

a function is uniquely determined by the images of the standard vectors in its domain (true false)

this is false. a function is only uniquely determined if the matrix of the images have a rank equal to the (number of rows)

a matrix transformation induced by a matrix A is a linear transformation (true false)

true

for any two matrices A and B, if AB=In, then A is invertible

true

if some row of a square matrix consists of only zero entries, then the determinant equals zero

true

a matrix is invertible if and only if its columns and rows are linearly independent

true (part of theorem 2.6)

the zero transformation is linear (true false)

true because this transformation preserves both properties

every linear transformation is a matrix transformation (true false)

true becuase a linear transformation is a matrix transformation induced by the standard matrix

the composition of linear transformations is a linear transformation (true false)

true theorem 2.12

a linear transformation with codomain Rm is onto if and only if the rank of its standard matrix equals m (true false)

true! a function is onto if rank of the standard matrix equals m (the number of rows side note: a function is onto if -every vector in Rm is an image -the range is all of Rm -if the columns for a generating set for its codomain Rm

every linear transformation is the matrix transformation induced by its standard matrix (true false)

true! a linear transformation is a matrix transformation induced by the standard matrix

the first column of a standard matrix of a linear transformation is the image of the first standard vector under the transformation (true false)

true! this is the definition of a standard matrix

elementary matrices are invertible (true or false)

true, every elementary matrix is invertible. Furthermore, the inverse of an elementary matrix is also an elementary matrix

every linear transformation preserves linear combinations (true false)

true, that is why the transformations are called linear

If T is a invertible linear transformation, the function T-1 is a linear transformation (true false)

true.

for any two matrices A and B, if AB=In, then BA=In (true or false)

true.

it is easier to prove something is a linear transformation because....

you can find the standard matrix and prove T(v)=Av