Linear Algebra XM1
A 2x2 has an inverse iff δ=ad-bc≠0 so the inverse of A is
(1/δ)[d -b] [-c a]
matrix multiplication the fancy formal way for (AB)ij
(AB)ij: (k=1 to n)∑Aik*Bkj so AB∈Mp,q (ith row of A)*(jth row of B)
properties of transpose
(A^T)^T=A (A±B)^T=A^T±B^T (cA^T)=c(A^T)
transpose of a matrix
(A^T)ij=Aji if A is mxn, then A^T is nxm
(AB)^1=
(B^1)(A^-1)
If A is a matrix and c,d are constants, d(cA)=(cd)A
(d(cA))ij = d(cA)ij = d(cAij) = (dc)Aij) = (cd)Aij = ((cd)A)ij
if a=cb and c>0 then theta is
0°
properties of nonsingular matrices
1) (A-¹)-¹=A 2) (Aⁿ)-¹=(A-¹)ⁿ=A-ⁿ 3) (AB)-¹=B-¹A-¹ 4) (A^T)-¹=(A-¹)^T prove these by just multiplying both sides of the equation together rank(A)=n
properties of matrix multiplication
1) 0A=0∈Mr,q if A∈Mp,q and 0∈Mr,p 2) IA=A 3) (B+C)A=BA+CA 4) (AB)C=A(BC) 5) transpose of (AB) is B^T*A^T
method for finding the inverse of a matrix that has one. So if A is square
1) augment A to be a nx2n matrix, whose first n columns form A itself and the remaining n columns form the In 2) convert this [A-¹|In] to rref 3) A side has to resemble I. If it can't, stop 4) If it does and is in rref, then it should resemble [In|A-¹]
a matrix is in reduced row echelon form iff
1) first nonzero entry in each row is 1 2) each successive row has its first nonzero entry in a later column 3) all entries above and below the first pivot entry of each row are 0 4) all full rows of zeroes are in the final rows of the matrix
if a=cb and c<0 then theta is
180°
the solution set of a system of equations is unchanged if
3) list the equations in a different order 2) one equation is multiplied by a nonzero constant 1) replace one equation by: the sum of it and any multiple of another equation
if a*b=0 then theta is
90° (orthogonal)
effect of row operations on matrix multiplication
A,B are matrices s.t. their product is defined 1) if R is any row operation, then R(AB)=(R(A))B 2) if R1,...,Rn are row operations, then Rn(...(R2(R1(AB)))...) = (Rn...(R2(R1(A)))...)B
AB=AC doesn't necessarily mean B=C. However, if A is invertible, then
A-¹(AB)=A-¹(AC) → B=C
dimension requirements for multiplying two matrices AB
AB can be multiplied iff Amn and Bnp resultant dimension is mp
If A is a square matrix, then B is an inverse to A iff
AB=I
definition of matrix inverse (invertible)
AB=I and BA=I B=A^-1 and A=B^-1
proof that if A,B are inverses of each other and A,C are inverses of each other, then C=B
AB=I, BA=I, AC=I, CA=I BAC=(BA)C=IC=C BAC=B(AC)=BI=B so C=B
homogeneous system and properties
AX=0 trivial solution is one with all zeroes, nontrival otherwise in rref, if Amn has fewer pivot entries than n, system has a nontrivial solution (rank<n) if A has exactly the same amount of pivot entries as n, then the system has only the trivial solution (rank=n) if there are fewer equations than variables, then there's at least one nonpivot column (with at least one independent variable taking on any value), meaning there would be infinite solutions
diagonal matrices
Aij=0 whenever i≠j has to be square
proof that (B+C)A=BA+CA
A∈Mp,q and B,C∈Mr,p ((B+C)A)ij = (k=1 to p)∑(B+C)ik*Akj = ∑(Bik+Cik)*Akj = ∑Bik*Akj+Cik*Akj = ∑Bik*Akj+∑Cik*Akj = (BA)ij+(CA)ij = (BA+CA)ij) //
proof that IA=A
A∈Mp,q and I∈Mp,p then IA=A∈Mpq the (ij) entry of IA is (k=1 to p)∑(I)ik*(A)kj
prove that if AC=B and [A|B]~[D|G] then DC=G
En...E1[A|B]=[D|G] En...E1*A=D and En...E1*B=G then DC=(En...E1*A)C=En...E1*(AC)=En...E1*B=G corollary: If X1 is a solution to AX=B then X1 is also a solution to DX=G. The converse is true
normalizing a vector
If x∈Rⁿ and x≠0, then u=(1/‖x‖)*x is a unit vector in the same direction of x
scalar multiplication of vector magnitude
Let x∈Rⁿ and c∈R, then ‖c‖*‖v‖=‖cv‖
singular matrix
a matrix that has no inverse
finding angle btwn two vectors using dot product
a*b=‖a‖‖b‖cosθ, where θ is the angle btwn a and b
length of vector
also called norm or magnitude if v=[v1,v2,...] then ‖v‖=√((v1)²+(v2)²+...) ‖v‖²=v*v or ‖v‖=√(v*v)
unit vector
any vector of length 1
proof that xy=‖x‖‖y‖ iff y=cx, c>0
assume that y=cx xy = x(cx) = c‖x‖² = c‖x‖‖x‖ = ‖x‖‖cx‖= ‖x‖‖y‖
proof by induction
base step: prove that the desired statement is true for the initial value i∈Z inductive step: assume that if the statement is true for an integer value k=i, then the statement is true for the next integer value k+1 as well
If A is a matrix and c is a constant, (cAij)=?
c(Aij)
scalar matrices
c*I (scalar multiplication of identity matrix)
matrix reflexivity
every matrix A is row equivalent to itself proof: IC=C
prove that if A,B are both square, of the same size, and invertible, then AB is also invertible
find an inverse for AB: (inverse)AB=AB(inverse)=I B^-1(A^-1(AB))=B^-1((A^-1*A)B)=B^-1(IB)=I do same for A
proof that row operations do not change solution sets
given M and B, let N=[M|B] if v is a solution s.t. Mv=B E(Mv)=EB → (EM)v=(EB) so [EM|EB] →E[M|B]
matrix transitive property
if A∼B and B∼C, then A∼C Proof: EnEn-₁...E1*A=B and Fm...F1*B=C So Fm...F1((En...E1)A)=C → (Fm...F1En...E1)A=C
matrix symmetry
if A∼B, then B∼A ∼ is an equivalence relation (think of E) proof: En...E1A=B, then (E^-1)n[En...E1A]=(E^-1)n[B] so En-1...E1A=(E^-1)nB. Now iterate until A=(E^-1)1...(E^-1)nB
geometric interpretation of vectors
if v∈R², imagine coordinates. v would be the movement from one point to another ex. (3,2) to (1,5) so v would be [-2,3]
finding individual solutions to a system that has infinite soutions
let nonpivot variables take on any real value and derive the other pivot variables from these choices
if A is a not a square matrix, then it has ___ inverse
no
rank of matrix
number of nonzero rows in row reduced echelon form two row equivalent matrices have the same rank if equations are consistent, solution set has (# of variables) - (rank) = degrees of freedom in answer = dimension of solution space less than or equal to number of rows
AX=B has an infinite number of solutions if
one of the columns left of the augmentation bar has no pivot entry in rref. The nonpivot columns correspond to (independent) variables that can take on any value, and the values of the remaining (dependent) variables are determined from those. at least two solutions
elementary matrices
result of performing one row operation on the identity matrix every elementary matrix has an elementary inverse
if A~B then they have the _____ solution set
same
linear combination
sum of scalar multiples of a list of vectors
symmetric and skew-symmetric matrices
symmetric: A=A^T; A+A^T skew-symmetric: A=-A^T; A-A^T
inconsistent system
system's solution set is empty; no solutions
find only a particular row or column of a matrix product
the kth row of AB is the product (kth row of A)*B the ith column of AB is the product A*(lth column of B)
row space
the subset of R^n consisting of all vectors that are linear combinations of the rows of A a question could be to determine whether a vector is in the row space of A, given dimensions matched ex. [5,2]=c1[1,2]+c2[2,1]+... 1c1+2c2=5 2c1+1c2=2
row equivalent
two matrices C,D are row equiv if there's a sequence of elementary matrices E1,E2,...,En s.t. En...E2E1C=D every matrix is row equivalent to a unique matrix in reduced row echelon form
parallel vectors
two vectors are parallel if they are in the same direction (x=cy) xy=±‖x‖‖y‖ and cosθ=±1 (θ=0° or 180°)
upper/lower triangular matrix
upper: Aij=0 if i>j lower: Aij=0 if i<j
what's a dot product (applications?)
vector multiplication (v*w=v1w1+v2w2+...∈R^n) angle between two vectors √(v*v)=‖v‖ (length/magnitude of vector v
proof that if a,b are unit vectors then -1 ≤ a*b ≤ 1
where does a*b show up in a dot product? it shows up in (a+b)(a+b) or (a-b)(a-b) so (a+b)(a+b) = ‖a+b‖² ≥ 0 (a*a)+2ab+(b*b) = ‖a‖²+2ab+‖b‖² = 1+2ab+1≥0 so ab≥-1 (a-b)(a-b) = ‖a-b‖² ≥ 0 ultimately ab≤1 //
orthogonal vectors
xy=0 x and y are perpendicular to each other
prove that if xy=±‖x‖ ‖y‖ (cosθ=±1) then x and y are parallel (y=cx)
y=cx → xy = x(cx) = c‖x‖² so c=xy/‖x‖² xy≠0 bc ‖x‖ and ‖y‖ are nonzero show that y-cx=0 or y-(xy/‖x‖²)x=0 using the property that zz=0 iff z=0, show that [y-(xy/‖x‖²)x]²=0 and we're good
Cauchy-Schwarz Inequality
|xy|≤‖x‖‖y‖
proof of reverse triangle inequality
|y|=|x+y-x|≤|y-x|+|x| |x|=|y+x-y|≤|x-y|+|y| |x+y-x|≤|y-x|+|x| → |x+y-x|-|x|≤|y-x| |y+x-y|≤|x-y|+|y| → |y+x-y|-|y|≤|x-y| so |y|-|x|≤|y-x| and |x|-|y|≤|x-y| |y-x|=|x-y| |x|-|y|=-(|y|-|x|) so |(|x|-|y|)|≤|x-y|
triangle inequality
‖x+y‖≤‖x‖+‖y‖
properties of matrix addition
∀ A, B, B ∈Mm,n 1) A+B=B+A 2) (A+B)+C=A+(B+C) 3) A+0=A 4) A+(-A)=(-A)+A=0 OR ∀ A∃B s.t. A+B=0 5) c(A+B)=cA+cB 6) (c+d)A=cA+dA 7) (cd)A=c(dA) 8) A+B=C+B then A=C
properties of scalar multiplication of matrices
∀ A, B, C ∈ Mm,n and c∈R 1) c(dA)=(cd)A 2) 1A=A 3) c(A+B)=cA+cB 4) (c+d)A=cA+dA
properties of vector multiplication
∀u,v,w∈Rⁿ Commutative: u*v=v*u Distributive: (u*v)w=uw*vw Zero Vector: If v*v≥0, v*v=0 → v=zero vector Associative: (r*u)v=r(u*v)
Properties of vector addition
∀u,v,w∈Rⁿ and ∀r,s∈R Associative: u+v=v+u Commutative: (u+v)+w=u+(v+w) Addition identity: v+zero vector=v Cancellation: u+w=v+w → u=v Distributive: (r+s)v=rv+sv AND ru+rv=r(u+v) Associative: (rs)v=r(sv) Multiplication identity: 1*v=v
