linear alg
If A is a matrix... A is not invertible if det(A)=
0
If v is a vector, Av=lambda v can be re-written as...
0= lambda v - Av 0= lambda In v - Av since v=In v lambda In v - Av = 0 (lambda In - A)v = 0 where lambda In - Av is some matrix B then Bv = 0
To diagonalize a matrix...
1) Since we are looking for vectors, if the vector can't be 0 then the det of A- lambda I must be 0. 2) Subtract lambda from the main diagonal entries for v in (A-lambda I)=0 3) Check that eigenvectors are linearly ind of each other 4) Post multiply matrix A with the matrix composed of these eigenvectors and pre-multiply A with the inverse of P such that D= P^-1AP (the inverse of P is equal to 1/detP * the transpose of the cofactor matrix) 5) the result is the eigenvectors in the diagonal of the matrix
What are the 2 conditions for a matrix to be invertible?
1) T has to be onto 2) T has to be 1:1
What are all of the row operations? (3)
1) swap 2 rows 2) Scale- multiply a row by a constant 3) pivot- add a multiple of one row to another
Finish the thm: Regarding W perp with W a subspace of R^n... 1) A vector x is in W perp if and only if x is orthogonal to every vector in a a set that spans W. 2)
2) W perp is a subspace of R^n
Determinant of a 2x2 matrix [ a b ] [ c d ]
= ad-bc
Finish the thm: An nxn matrix is said to be orthogonally diagonizeable if there are an orthogonal matrix P (with P^-1 = P transpose) and a diagonal matrix D such that...
A = PDPtranspose = PDP^-1
What is a basis for a subspace, say H?
A basis for a subspace H of R^n is a linearly independent set in H that spans H.
What is a free variable variable? How many free variables are in a 3x5 augmented matrix?
A free variable is a column with no pivot. If you have m unknowns and n independent and consistent equations, where n≤m, you have m-n free variables. In a 3x5 augmented matrix, you have 4 variables and only 3 independent equations, so one variable is free (you have to reduce the matrix to row echelon form in order to know which variable is free).
Definition of a homogeneous equation
A linear system of equations Ax=b is called homogeneous if b=0 (x=0 is always a solution of the homogeneous equation)
In order for the transformation x->Ax to be 1:1...
A must have a pivot in every column
What is a Markov chain?
A series of discrete time intervals over which a population distribution at a given time (t = n; n = 0,1,2, ... ) can be calculated based on the the distribution at an earlier time (t = n-l) and the probabilities governing the population changes. More specifically, a future distribution depends only on the most recent previous distribution. A Markov chain specifically is a sequence of probability vectors x0, x1, x2..., together with a stochastic matrix P such that x1= Px0, x2= Px1, x3= Px2 ... Thus the Markov chain can be described by the first order difference equation x(k+1) = Pxk
What is a stochastic matrix?
A square matrix whose columns are probability vectors
Unit vector def
A vector whose length is 1
What is a probability vector?
A vector with non-negative entries that add up to 1
If B=A^-1, then...
A=B-1. In other words, if A is invertible , so is A^-1 and A=(A^-1)^-1.
Finish the thm. If A and B are invertible then...
AB is also invertible and (AB)-1= B^-1A^-1
If H is a subspace of R^3, then there is a 3x3 matrix such that H=
ColA
Finish the thm: If a set contains more vectors than there are entries in each vector, then the set is linearly ________.
Dependent. That is, any set {v1, ..., vn} in R^m is linearly dependent if m<n(or equal to). There are more variables than there are equations so there must be a free variable.
For any eigenvalue, lambda, the eigenvectors that correspond to that lambda is the eigenspace Elambda which is written...
Elambda= Nul(lambda In - A) The null space of this matrix is the set of all the vectors that satisfy 0=(lambda In -A)v
Def of Span
For a nonempty set S = {u1, . . . , uk} of vectors in R^n, the span of S is the set of all linear combinations of u1, . . ., uk in R^n (in other words, ALL vectors of the form c1 u1 + · · · + ck uk for any choice of coefficients). This span is denoted span S = span{u1, . . ., uk}.
Explain ||cv|| = |c|||v|| where c is a scalar and v is a vector
For any scalar c, the length of cv is |c| times the length of v
What is the distance between u and v, written as dist(u,v)?
For u and v in R^n, the distance between u and v is the length of the vector u-v. That is, dist(u,v)= ||u-v||
What is a coordinate vector?
Given a vector x and a basis B= {b1,...,bn}, the coordinate vector x in terms of B (called [x]B) is the coefficients that go in front of the basis vectors in B to get to x
What is the transpose of a matrix?
Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by AT, whose columns are formed by the corresponding rows of A
The diagonalization of an nxn matrix A such that A = PDPtranspose = PDP^-1 requires n linearly independent and orthonormal eigenvectors. When is this possible?
If A is orthogonally diagonizable as shown then Atranspose= (PDPtranspose)transpose = (Ptranspose)transposeDtransposePtranspose= PDPtranspose=A. Thus A is symmetric.
What is the inverse matrix? What are two ways you can compute it?
Let Mn(R) denote the set of all nxn matrices with real entries. Let A be in Mn(R). Suppose there exists an nxn matrix B such that AB=BA=In. Then the matrix is called invertible and B is called the inverse of A (denoted A^-1). 1)Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A^-1 ]. 2) A^-1 = 1/detA * (cofactor matrix of A)transpose
What is the dimension of the column space?
The dimension is just the number of vectors in any basis. All bases have the same number of vectors for any given subspace.
Def of a non-trivial solution
The not-so-obvious solution, having some variables or terms that are not equal to zero. A set of vectors is linearly dependent, that is if one of the vectors is some combination of the others.
Span of 0?
The span of 0 is the zero vector {0}. Note that the span of any other single vector consists of infinitely many vectors (all multiples of the fixed one).
def of a linear transformation
The transformation T applied to some vector x is equal to some matrix times x that is an mxn matrix T(x, a vector)=Ax
Transpose of a 2x2 matrix [ a b ] [ c d ]
[ a c ] [ b d ]
What is the inverse of the 2x2 matrix [ a b ] [ c d ]
[ d -b ] [ -c a ]
An orthogonal basis for a subspace W of R^n is...
a basis for W that is also an orthogonal set.
What is a symmetric matrix?
a matrix A such that AtransposeT=A, which requires that it is square. It's main diagonal entries are arbitrary, but its other entries occur in pairs- on opposite sides of the main diagonal.
Definition of a non-homogeneous equation
b does not equal zero
characteristic polynomial formula
det(A- lambdaI)
What does it mean for a transformation to be "onto"?
for any vector b that is a member of R^m there exists at least one solution to Ax=b where x is a vector that is a member of R^n
The Euclidean space R^n, where the inner product is given by the dot product...
if a and b are vectors... a dot b = ax*bx + ay*by =|a||b|cos(theta)
Define the length of a vector
if a is a vector... ||a||= sqrt(a1^2 + a2^2 + a3^2 + ... +an^2) = sqrt(a dot a) or ||a||^2= a dot a
If a pair of vectors are linearly _____, then they are not multiples of eachother.
independent
A set {u1,...,up} is an orthonormal set if
it is an orthogonal set of unit vectors.
What is the identity matrix?
it is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In
If A is mxn and the linear transformation x->A is onto, then rankA=
m
If there are some non-zero vector v's that satisfy Av=lambda v then the matrix lambda In - A...
must have a determinant of 0. Av= lambda v for any non-zero v's if and only if the det(lamda In - A) = 0. In other words, lambda is an eigenvalue of A iff det(lamda In - A) = 0
rankA + dimNulA=n where n is the number of columns in A. As the number of columns of A is n, and rankA=k, dimNulA=
n-k
Matrices are row equivalent if:
one can be changed to the other by a sequence of elementary row operations
Finish the thm: If A is symmetric, then any two eigenvectors from different eigenspaces are...
orthogonal
The set of all vectors z that are orthogonal to W is called the
orthogonal complement of W and is denoted by W perp
If a vector z is orthogonal to every vector in a subspace W of R^n , then z is said to be
orthogonal to W
An mxn matrix U has...
orthonormal columns if and only if UtransposeU = I
If {v1, v2, . . ., vn} is a basis for a vector space V, then any vector v ∈ V has a unique representation v = x1v1 + x2v2 + · · · + xnvn, where xi ∈ R. The coefficients x1, x2, . . ., xn are called the
the coordinates of v with respect to the ordered basis v1, v2, . . ., vn.
The rank of a matrix equals...
the dimension of it's row space. Specifically, the rank of a matrix is defined as a) the max # of linearly independent column vectors in the matrix or b) the max number of linearly independent row vectors.
If A is row equivalent to the identity matrix, then A is invertible, because
the identity matrix is invertible
The dimension of a vector space is...
the number of coordinates you need to describe a point in it. Thus a plane in R^3 is of dimension 2 since each point in the plane can be described by 2 parameters even though the actual point will be of the form (x,y,z)
For T to be onto...
the span of the column vectors must be equal to R^m or in other terms, col(a)=R^m
The row vectors of A are the vectors in R^n corresponding to the rows of A. The row space of A is
the subspace of R^n spanned by the row vectors of A.
The row space of matrix A equals the columns space of
the transpose of A
When looking for eigenvalues, you can assume that...
the vector v does not equal 0
Finish the thm: If a set S={v1, v2, v3,...vn} in R^n contains _______, then the set is linearly dependent.
the zero vector
If S= {u1,...,up} is an orthogonal set of nonzero vectors in R^n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
Every equation Ax= 0 has the ______ solution whether or not some variables are free.
trivial
In which cases can a set span R^m and in which cases can it not?
when a matrix is mxm or m<n, the matrix spans R^m; it doesn't span R^m when m>n
The Leontief Input-Output Model can be described by the equation...
x = Cx + d where x is the production matrix (amount produced), C is the input-output matrix (intermediate demand) and d is the consumer demand matrix (final demand). The production matrix can be solved by the formula: Ix - Cx=d or (I-C)x=d
Let U be an mxn matrix with orthonormal columns, and let x and y be in R^n. Then a.) ||Ux|| = ||x|| b.) (Ux) dot (Uy)= x dot y c.) (Ux) dot (Uy) = 0 if and only if x dot y = 0 What do properties a and c say about the linear mapping x->Ux ?
x-> Ux preserves lengths and orthogonality.
Does order matter in matrix multiplication?
yes
Let V be a vector space. A function α : V → R is called a norm on V if it has the following properties: Notation. The norm of a vector x ∈ V is usually denoted ||x||
(i) α(x) ≥ 0, α(x) = 0 only for x = 0 (positivity) (ii) α(rx) = |r| α(x) for all r ∈ R (homogeneity) (iii) α(x + y) ≤ α(x) + α(y) (triangle inequality)
What are the 3 types of solutions to a matrix system?
0 (inconsistent), 1 (no free variables and consistent), or infinitely many solutions (at least one row is all zeroes-dependent system of eqs)
What is a diagonal matrix?
A square matrix in which all non-diagonal entries are zeroes
Def of a consistent system
A system of linear equations is consistent if it has at least one solution.
Given a matrix A and a solution set b, how do you determine if it is consistent?
Determine if b is a linear combo of the column vectors of matrix A, which is the same as determining if b is in span {v1...vn}. b is in the span{v1...vn} if there is a pivot in every row.
True or False: In some cases, it is possible for 4 vectors to span R^5.
False. If a set {v1,v2,v3,v4} were to span R^5, then the matrix A=[v1 v2 v3 v4] would have to have position in each of its 5 rows, which is impossible since A has only 4 columns.
How do you find the cofactor matrix?
In the case of a 3x3 matrix, for each position, take the 2x2 determinant of the matrix that exists when you cover up the row and column you are on by evaluating ad-bc. There are predetermined - and + positions which looks like [ + - + ] [ - + - ] [ + - + ]
What is the null space?
The null space of a matrix A is the set Nul A of all the solutions of the homogenous equation Ax=0. It is a subspace of R^n. In other words, the null space is the set of all vectors that "loose their identity" as h is applied to them in the equation h(v)= A*v
What is an eigenvalue?
There are vectors that get scaled up or down by a transformation by some factor, lambda, the eigenvalue. t(v)=lambda v= Av
Def Identity Matrix
This matrix, denoted I , is a square matrix (nxn). When any m×n matrix is multiplied on the left by an m×m identity matrix, or on the right by an n×n identity matrix, the m×n matrix does not change.
True or False: If {u, v, w} is linearly dependent, then u,v, and w are not in R^2.
True. To be linearly dependent: m<n (or equal to) 3 vectors in the set and to entries (R^2), so 2<3. Therefore u,v, and w have to be in R^2.
T or F: An eigenvector cannot be 0, but an eigenvalue can be 0.
True. Therefore, there must be some nontrivial vector x for which Ax=0x = 0
Regarding LU factorization, how do you find L and U?
U is just the upper triangular matrix of the matrix in reduced row echelon form. To get L, start with a matrix a lower triangular matrix with unknowns in the lower part. To get the unknown in the i,j position, take whatever number was used to eliminate the entry in the same column of the number and divide it by the original number in the i,j position.
If Ly=b and Ux=y, then...
[L|b] =[I|y] and [U|y]=[I|x]
What is a change of coordinates matrix? Given B= { [1] , [ 1] [1] [-1]} and C= { [2] , [1] [3] [0] } How would you find Pc-> B?
a) Given a bases B and C, the change of coordinates matrix, from B to C, called Pc<-B is the matrix that changes coordinate vectors [x]B to [x]C, that is Pc <- B[x]B=[x]C. The columns of Pc<-B are the C coordinate vectors of B: [b1]c...[bn]c. b) row reduce the augmented matrix: [ 2 1 1 1 ] [ 3 0 1 -1 ] Pc<-B will be the 2x2 matrix on the right [1/3 -1/3] [1/3 5/3]
In an mxn matrix, if m<n, then the max rank of the matrix is __. If m>n, then the max rank of the matrix is __.
a) m b) n
If AB=BA, we say that matrices A and B...
commute
The max number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply...
transform the matrix to its row echelon form and count the number of non-zero rows.
Two vectors u and v in R^n are orthogonal (to each other) if...
u dot v = 0 This means that two vectors are geometrically perpendicular if and only if the distance from u to v is the same as the distance from u to -v, which is the same as requiring the squares of the distances to be the same. [dist(u, -v)]^2 = ||u - (-v)||^2 = ||u + v||^2 = ||u||^2 + ||v||^2 + 2u dot v and... [dist(u, v)]^2 = ||u ||^2 + ||-v||^2 + 2u dot (-v)= ||u||^2 + ||v||^2 - 2u dot v Therefore, two squared distances are equal if and only if 2u dot v = -2u dot v which happens if and only if u dot v= 0
If we divide a nonzero vecotr v by its length- that is, multiply by 1/||v||- we obtain a
unit vector u because the length of u is (1/||v|| )||v||. The process of creating u from v is called normalizing v and we say that u is in the same direction as v.
If the vector u is a member of the orthogonal compliment of the row space, u is orthogonal to any member of your row space, which means that u dot any of the row vectors equals...
zero. so u dot r1=0, u dot r2=0, u dot r3=0 then this implies Au=0, which means, that u is a member of the null space
How do you compute the determinant of a matrix great than a 2x2 matrix, say a 3x3 matrix?
Firstly, you can only take the determinant of a square matrix. A= [ a b c ] [ d e f ] [ g h i ] 1) Choose a column or row to expand upon (choose ones with a lot of zeroes or ones to make the calculation easier). Let's expand on column a. 2) Moving down/across the column/row... cover up the first column if you are expanding on a row or cover up the first row if you are are expanding on a column. In this case we cover up row a. 3) The entry where the column/row you expanding on intersects the of the row/column you are covering up is your first coefficient, which is a in this case. 4) Take the determinant of the 2x2 matrix that is left when you cover up the column/row you are expanding on and the row/column you are covering up using the formula... Determinant of a 2x2 matrix [ a b ] [ c d ] = ad-bc so we would need to compute ei-hf 5) Keep doing this for each row/column that you cover up as you move across/down 6) Then we add and subtract the resulting terms, alternating signs (add the $ a$-term, subtract the $ b$-term, add the $ c$-term.) 7) detA = a[ e f ] -d [ b c ] + g[ b c ] [ h i ] [ h i ] [ e f ]
The set span{v1, v2, ..., vp} contains all linear combinations of v1, v2, ..., vp implies...
a) it is closed under addition b) it is closed under scalar multiplication c) the zero vector of v is also in span{v1, v2, ..., vp} as 0=0v1 + 0v2 + ... + 0vp So, the span{v1, v2, ..., vp} is a subspace of V. Any subspacce of a vector space is itself a vector space.
What is the invertible matrix thm?
The nxn matrix A is invertible if and only if 0 is not an eigenvalue of A. (If 0 is an eigenvalue of A, then A is invertible)
Def of a trivial solution
The obvious solution, the trivial solution is the unique solution where every variable equals 0. If a system of equations has only the trivial solution, then it is linearly independent.
Suppose that T is a rotation in R2, T rotates through angle a (all rotations are counterclockwise). The standard matrix of T is...
[ cos(a) -sin(a) ] [ sin(a) cos(a) ]
Finish the thm: Let V be a vector space over the field k and let W be a subset of V. Then W is a subspace if and only if W satisfies three conditions...
1) The zero vector, 0, is in W 2) If u and v are elements of W, then the sum u+v is an element of W 3) If u is an element of W, and c is a scalar from k, then the product cu is an element of W
The dot product/ inner product satisfies the following four properties. Let u, v, and w be vectors and alpha be a scalar, then:
1. (u+v dot w)=(u dot w)+(v dot w). 2. (alphav dot w)=alpha(v dot w). 3. (v dot w)=(w dot Explainv) (commutative property) 4. (v,v) >=0 and equal if and only if v=0. (positive definite condition)
Def of spanning set
A subset S of a vector space V is called a spanning set for V if Span(S) = V. ex: Vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1) form a spanning set for R^3 as (x, y, z) = xe1 + ye2 + ze3.
What is an eigenvector?
Any vector v that satisfies the equation for the transformation, T. It is scaled up or down by an eigenvalue during a transformation. t(v)=lambda v= Av
What dies it mean for a function to be one-to-one?
At most there is only 1 x that maps to a y in the range. The function is not one-to-one if 2 elements of x map to the same y
In the true statement, "A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in Rm. Such a system Ax = 0 always has at least one solution, namely x = 0 (the zero vector in R^n)," what does x in R^m and R^n" mean?
If A has m rows and n columns, then x must be a vector with n elements and b must be a vector with m elements. Otherwise, the dimensions are not compatible and the matrix multiplication doesn't make any sense.
v1= [1], v2= [0], v3= [1] [0] [-1] [0] [-1] [0] [0] [0] [1] [-1] Does {v1, v2, v3} span R^4?
No. Let v1, v2, v3 be the columns of the matrix. If A has a pivot position in every row, then its columns will span R^4. In this case, there are only three columns in it so there can be at most 3 pivot positions.
Let A be an mxn matrix. The orthogonal complement of the row space of A is _____ , and the orthogonal complement of the column space of a A is _____.
a) the null space of A b) the null space of A transpose That is (RowA) perp = NulA and (ColA)perp = NulA transpose REMEMBER: if x is a vector, NulA = {x in R^n | Ax=0} So if you have any vector that is a linear combo of the row vectors (A transpose) and you dot it with any member of your null space you will get 0.
row echelon form if:
a.) all non zero rows are above rows of all zeroes b) the leading coefficient is always to the right of the leading coefficient of the row above it
reduced row echelon form if:
a.) it is in row echelon form b.) every leading coefficient is 1 and is the only non zero entry in its column
If W is the subspace spanned by an orthonormal set, then {u1,...,up} is...
an orthonormal basis since the set it is automatically linearly independent. To show that a set is an orthonormal basis, compute the dot product of each vector to see if it equals zero.
Finish the thm: a) If A has n linearly independent eigenvectors p1,p2,...pn with corresponding eigenvalues lambda1, lambda2, ..., lambdan, and if P is a matrix with columns p1, p2, ...,pn then D= P^-1 AP b)
b) If D= P^-1AP=D for some diagonal matrix D with entries lambda1, lambda2, ..., lambdan down the main diagonal, then the columns of P are linearly independent and are eigenvectors of A with corresponding eigenvalues lambda1, lambda2, ..., lambdan
Finish the thm: Let A be an mxn matrix. The following statements are either all True or all False: a) for each b in R^m, the equation Ax=b has a solution b)Each b in R^m is a linear combo of the columns of A c)? d)?
c)The columns of A span R^n d)A has a pivot position in every row
Let {u1, ..., up} be an orthogonal basis for a subspace W of R^n. For each y in W, the weights in the linear combination y= c1u1 + ... + cpup are given by...
cj = (y dot uj)/(uj dot uj) where j= 1, ..., p
The diagonal subdivides the matrix into two blocks, one above the diagonal and below it. If the lower block consists of all zeroes, we call such a matrix _____. If the upper block consists of all zeroes, we call such a matrix ____.
lower triangular upper triangular
If x is a vector... ProjL(x)=
some vector v in line L where x-ProjL(x) is orthogonal to L. This implies that the projection is some scalar c*v since some vector v in line L is a a scalar of v. So we can rewrite x-ProjL(x) as x - cv. Since x-cv is orthogonal to L, by definition, the dot product of x-cv with any vector in L is 0. Therefore (x-cv) dot v= 0 and we can solve for c. c = (x dot v)/ (v dot v) Then we can write ProjL(x)= ((x dot v)/ (v dot v))v
You can only multiply a matrix when
the number of columns in A = the number of rows in B 2A3 x 3B2 = 2C2
What is the image of a linear transformation
the span of the vectors of the linear transformation. (Think of it as what vectors you can get from applying the linear transformation or multiplying the matrix by a vector.) It can be written as Im(A).
If there are more rows than columns then you can't have a pivot in each column, which means...
there is a free variable and Ax=0 can have infinitely many solutions (linearly dependent)
What is the general formula for discrete dynamical systems?
x_(k+1)= A_x_k This gives the long term behavior of a system. Vectors xk give information about the system denoted by k passes. Since {v1, ..., vn} is a basis for R^n, any initial vector x0 can be written uniquely as x0= c1v1 + ... + cnvn. The eigenvector decomposition of x0 determines what happens to the sequence {xk}. Since the vi are eigenvectors, xi =Ax0 = c1Av1 + ... + cnAvn = c1 lambda1 v1 + ... +cnAvn In general xk = c1(lambda1)^k v1 + ... + cn(lambdan)^k vn