Linear Algebra
A transformation or mapping of T is linear IF
(i) T (u + v) = T(u) + T(v) for all u, v in the domain of T; (ii) T (cu) = cT(u) for all u and all scalars c
row reduction algorithm
1) Locate the first pivot column/first pivot position 2) Locate a pivot, choose a nonzero entry in the pivot column to be the pivot, if necessary do row interchange to move the pivot to the pivot position, if necessary do row scaling to make the pivot equal to 1 3) Create zeros below the pivot use row replacement to make all entries below pivot equal to 0 4) Cover the row containing the pivot and any row above it, apply steps 1-3 to the remaining submatrix 5) Start from the right most pivot and then move upward and to the left, if a pivot does not equal 1, make it equal 1 with row scaling, use row replacement to create zeros above each pivot
Properties of Matrix Multiplication
1. A(BC) = AB(C) 2.A(B+C) = AB +AC (A+B)C = AC+BC 3. r(AB) = (rA)B = A(rB) 4. if A (mxn) ImA = A = AIn *not commutative! *no canceling! * if AB=0, neither A nor B could necessarily equal 0
echelon form
1. All nonzero rows are above any all zero rows; 2. Each leading entry is in a column to the right of the previous leading entry; 3. All entries below a leading entry in its column are zeros
vectors in R3
3 x 1 column matrices with three entires. in 3d space geometrically
A transpose transpose
A
Let A, B, and C be matrices of the same size, and let r and s be scalars.
A + B = B + A (A+B)+C = A+(B+C) A+0=A r(A+B)=rA+rB (r+s)A = rA+sA r(sA) = (rs)A
pivot column
A column that contains a pivot position
general solution
A family of solutions that contains all possible solutions of a linear system.
each column of AB
A linear combination of the columns of A using weights from the corresponding column of B
identify matrix
A square matrix that, when multiplied by another matrix, equals that some matrix.
identity matrix
A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else.
linear combination
A sum of scalar multiples of vectors. The scalars are called the weights.
A+B transpose
A transpose + B transpose
if A is men matrix and u and v are vectors in Rn and c is scalar
A9u+v) = Au+Av and A(cu)=c(Au)
Characterization of Linearly Dependent Sets
An indexed set S = {v1,...,vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others.
Homogenous Linear System
Ax = 0
matrix equation
Ax=b
AB Transpose
B transpose x A transpose (reverse order!)
matrix transformation domain and codomain
Domain of T is Rn when A has n columns and codomain of T is Rm when each column of A has m entries.
A set of one vector is linearly independent
IFF the vector V is not the zero vector
Row-Column Rule for Computing AB
If the product AB is defined, then the entry in row I and column j of AB is the sum of the products of corresponding entries from row I of A and column j of B.
existence question
Is the system consistent?
uniqueness question
Is there only one solution?
The set of all vectors with two entries is denoted by
R 2 (R stands for real numbers and 2 indicates each vector contains two entires)
matrix transformation range
Range of T is set of all linear combinations of Columns of A because each image T(x) is of the form Ax
Elementary Row Operations
Replacement, Interchange, Scaling
Reduced Echelon Form
Same as echelon form, except all leading entries are 1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
geometric description of Span {v}
Span of {v} is set of all scalar multiples of v, which is set of points on line in R3 through V and 0
Transpose of a Matrix
Switch the rows and columns - imagine it kinda swinging up/down a 90 degree angle
Let T:Rn-->Rm be a linear transformation and A is standard matrix for T, then
T maps Rn onto Rm IFF columns of A span Rm T is 1-1 IFF columns of A are linearly independent
If T is a linear transformation, then
T(0) = 0 and T(cu + dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.
Th10: let Rn to Rm be a linear transformation, then there exists a unique matrix transformation such that
T(x) = Ax for all x in Rn. A is the men matrix whose jth column is the vector T(ej) and ej is the jth column of identity matrix in Rn
Linear independence of matrix columns
The columns of a matrix A are linearly independent IFF the equation Ax = 0 has only the trivial solution.
standard matrix
The matrix A such that T(x) = Ax for all <x> in the domain of T.
Codomain of T
The set R^m
domain of T
The set R^n
if a set contains more vectors than there are entries in each vector
Then the set is linearly dependent. (columns (p) greater than rows (n).
augmented matrix
a coefficient matrix with an extra column containing the constant terms
linear dependence relation
a homogeneous vector equation where the weights are all specified and at least one weight is nonzero
Existence and Uniqueness Theorem
a linear system is consistent if and only if the rightmost column of the augmented matrix is NOT a pivot column - that is, if and only if an echelon form of the augmented matrix has NO row of the form [0 ... 0 b] with b nonzero. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.
consistent linear system
a linear system with at least one solution
pivot position
a location in matrix A that corresponds to a leading 1 in the reduced echelon form of A
onto
a mapping T: R^n-->R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n
matrix transformation
a mapping x |-> Ax where A is an m x n matrix and x represents any vector in Rn.
coefficient matrix
a matrix that contains only the coefficients of a system of equations
column vector (vector)
a matrix with only one column
nontrivial solution
a nonzero vector x that satisfies Ax=0,IFF the equation has at least one free variable
nonzero row or column
a row or column that contains at least one nonzero entry
asking whether vector b is in span
amounts to asking whether vector equation x1v1+x2v2...=b which is asking whether linear system with augmented matrix [v1...vp b] has a solution
linearly independent
an indexed set {v1, ..., vp} with the property that the vector equation x1v1 + x2v2 + ... + xnvn = 0 has only the trivial solution
free variable
any variable in a linear system that is not a basic variable
Ax=b has same soln set
as vector equation x1a1...= b which has same soln set as augmented matrix [a1...a2...b]
A set of two vectors is linearly dependent if
at least one of the vectors is a multiple of the other
the equation Ax=b has solution IFF
b is a linear combination of the columns of A, columns of A span Rm, A has pivot position in every row
a mapping T: Rn to Rm is said to be 1-1 if
each b in Rm is the image of at most one X in Rn.
linear equation in variables x1...xn
equation that can be written as a1x1+...+anxn = b
Is T 1-1? vs Does T map Rn onto Rm
existence question vs uniqueness
Geometric Description of R2
identify a geometric point (a, b) with column vector [a over b]. Regard R2 as the set of all points in the plane.
vectors in RN
if n is positive integer, RN denotes collection of all lists of n real numbers, usually written as nx1 column matrices.
Row-Vector Rule for Computing Ax
if product of Ax is defined then the I-th entries is Ax is sum of products of corresponding entries from row I of A and from vector x
row equivalent
if there is a sequence of elementary row operations that transforms one matrix into the other. if augmented matrices of two linear systems are row equivalent, then the two systems have same soln set
geometric description of span {u,v}
if v not multiple of u, then span {u,v}} is the plane in R3 that contains u, v and 0.
for x in Rn, the vector T(x) in R^m is the
image of x
Every matrix transformation
is a linear transformation
Transformation (function or mapping) T from Rn to Rm
is a rule that assigns to each vector x in Rn, a vector T(x) in Rm.
if A is an min matrix with columns a1...an and if x is in Rn, then product of A and X (Ax)
is linear combination of the columns of A using corresponding entries in x as weights.
leading entry
leftmost nonzero entry in a nonzero row
equivalent linear systems
linear systems that have the same solution set
size of matrix
m rows x n columns
AB has the same...
number of rows as A and same number of columns as B
T is onto Rm when the range
of T is all of the codomain Rm (T maps onto Rm if for each b in codomain Rm there exists at least one solution of T9x) = b).
types of solutions in linear system
one solution (intersect), infinite soln(same line), no soln (parallel lines)
vector
ordered list of numbers
Set of all images T(x)
range of T
span {v1...vp}
set of all linear combinations of v1...vp. set of all vectors that can be written in the form c1v1+c2v2....+cpvp
solution set of linear system
set of all possible solutions
Parametic vector equation
suppose Ax=b is consistent for some b and p is a solution. Then the solution set of Ax=b is set of all vectors of form w=p+v(h) where V9h) is any solution of the homogenous equation Ax=0
Let T:Rn-->Rm be a linear transformation. Then T is one-to-one IFF
the equation T(x) = 0 has only trivial solution
Ax is only defined if
the number of columns in A equals the number of entries in x
linearly dependent set
the set {v1, ..., vp} with the property that the vector equation c1v1 + c2v2 + ... + cnvn = 0 has weights c1...cp that are not all zero.
trivial solution
the solution x=0 of a homogeneous equation Ax=0
shear transformation
the transformation T:R^2 --> R^2 defined by T(x) = Ax
scalar multiple of u by c is
the vector cu obtained by multiplying each entry in u by c
zero vector
the vector whose entries are all zero
If a set contains the zero vector
then the set is linearly dependent
intro to linear transformation
think of Matrix A as an object that acts of vector X by multiplication to produce a new vector Ax: correspondence from x to Ax is a function from one set of vectors to another.
basic variables
variables corresponding to pivot columns in a matrix
vector equation and matrix
vector equation x1a1+..=b has same soln set as linear system whose augmented matrix is [a1,a2..b]
linear combination and matrix
vectors a1,a2, and b are columns of augmented matrix. to determine if b can be written as a linear combination of b, find that weights x1,x2 exist. this can be done using row reduction on matrix that corresponds to vector equation
contraction
when T: R2->R2 by T(x)=rx, and r is between or equal to 0 and 1
dilation
when T: R2->R2 by T(x)=rx, and r is greater than 1
when does mapping T not 1-1
when some b in Rm is image of more than one vector in Rn
when does T not map Rn onto Rm
when there is some b in Rn for which the equation T(x) = b has no soln
parametric description of a solution set
write basic variables in terms of free variables. free variables act as parameters
vector equation
x1a1 + x2a2 + ... + xnan = b
