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Linear Algebra Cheat Sheet (DRAFT) by

A cheat sheet for linear algebra midterms

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Linear Systems

Solution
Can have no solution, one solution or infinitly many. The solution is the inters­erction
Solution set
The set of all possible solutions
Consis­tency
A system is consistent if there is at least one solution otherwise it is incons­istent
Equivalent
Linear systems are equivalent if they have the same solution set
Row operations
Replac­ement, interc­hange and scaling
Row equivalent
If there is a sequence of row operations between two linear systems then the systems are row equivalent. Systems that are row equivalent has the same solution set.
Existence
If a system has a solution (i.e. consistent)
Uniqueness
Is the solution unique
Homogenous
A system is homogenous if it can be written in the form Ax = 0
Trivial solution
If a system only has a the solution x = 0. A system with no free variable only have the trivial solution.
Non-tr­ivial solution
A nonzero vector that satisfies Ax = 0. Has free variable.

Inverser of a Matrix

C is invertible if CA = In and AC = In
If A is (2x2) then, A-1 =
(A-1)-1 = A
(AB)-1= B-1A-1
(AT)-1= (A-1)T

Linear Transf­orm­ations

Tranfo­rma­tio­n/m­apping
T(x) from Rn to Rm
Image
For x in Rnthe vector T(x) in Rmis called the image
Range
The set of all images of the vectors in the domain of T(x)
Criterion for a transf­orm­ation to be linear
1. T(u + v) = T(u) + T(v)
2. T(cU) = cT(U)
Standard Matrix
The matrix A for a linear transf­orm­ation T, that satisfies T(x) = Ax for all x in Rn
Onto
A mapping T is said to be onto if each b in the codomain is the image of at least one x in the domain. Range = Codomain. Solution existance. ColA must match codomain.
One-to-one
If each b in the codomain is only the image at most one x in the domain. Solution Unique­ness.
 
T is one-to-one if and only if the cols of A are linearly indepe­ndent
Free variable?
If the system has a free variable, then the system is not one-to­-one. I.e. the homogenous system only has the trivial solution
Pivot in every row?
Then T is onto
Pivot in every column?
Then T is one-to-one
To determine whether a vector c is in the range of a T. Solution: Let T(x) = Ax. Solve the matrix equation Ax = c. If the system is consistent, then c is in the range of T.
  

The Invertible Matrix Theorem

The following statements are equivalent i.e. either they are all true or all false. Let A be a (nxn) matrix
𝐴 is an invertible matrix.
𝐴 is row equivalent to the 𝑛×𝑛 identity matrix
𝐴 has 𝑛 pivot positions.
The equation 𝐴𝐱 = 𝟎 has only the trivial solution
The columns of 𝐴 form a linearly indepe­ndent set
The linear transf­orm­ation 𝐱 ↦ 𝐴𝐱 is one-to­-one.
The equation 𝐴𝐱 = 𝐛 has at least one solution for each 𝐛 in Rn
The columns of 𝐴 span Rn
The linear transf­orm­ation 𝐱 ↦ 𝐴𝐱 maps Rn onto Rn
There is an 𝑛×𝑛 matrix 𝐶 such that 𝐶𝐴 = 𝐼.
There is an 𝑛×𝑛 matrix 𝐷 such that 𝐴𝐷 = 𝐼.
𝐴T is an invertible matrix.
The columns of A form a basis of R^n
Col A = R^n
Dim Col A = n
Rank A = n
Nul A = {0}
Dim Nul A = 0
The number 0 is not an eigenvalue of A
The determ­inant of A is not 0

Elementary Matrices

Elementary Matrix Is obtained by performing a single elementary row operation on an identity matrix
Each elementary matrix E is invertible
A nxn matrix A is invertible if and only if A is row equivalent to In.
A = E-1In and A-1 = EIn= In
Row reduce the augmented matrix [ A I ] to [ I A-1]
NOTE If A is not row equivalent to I then A is not invertible

Linear Indepe­ndence

A set of vectors are linearly indepe­ndent if they cannot be created by any linear combin­ations of earlier vectors in the set.
If a set of vectors are linear indepe­ndent, then the solution is unique
If the vector equeation c1v1 + c2v2 + ... + cp*vp = 0 only has a trivial solution the set of vectors are linearly indepe­ndent
Theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent
Theorem: If a set of vectors containt the zero vector, then the set is linearly dependent

Algebraic properties of a matrix

Matrix and vector sum
A(u + v) = Au + Av
Matrix, vector and scalar
A(cu) = c(Au)
Associ­ative law
A(BC) = (AB)C
Left distri­butive law
A (B + C) = AB + AC
Right distri­butive law
(B + C) = BA + BC
Scalar multip­lic­ation
r(AB) = (rA)B = A(rB)
Identity matrix multi
ImA = A = AIn
Commute
If AB = BA then we say that A and B commute with each others
 
(AT)T = A
 
(A + B)T = AT + BT
 
(AB)T = BT AT
 
For any scalar r, (rA)T = rAT
 

LU Factor­ization

Factor­ization of a matrix A is an equation that expresses A as a product of two or more matrices:
Synthesis: BC = A
Analysis: A= BC
Assump­tion: A is a mxn matrix that can be row reduced without interc­hanges
L: is a mxm unit lower triangular with 1's on the diagonal
U: is a mxn echelon form of A
U is equal to E*A = U, why A = E-1U = LU where L = E-1
See figure ** for how to find L and U
Find x by first solving Ly = b and then solving Ux = y

Row Reduction and Echelon forms

Leading entry
A leading entry refers to the leftmost non-zero entry in a row
Echelon form
Row equivalent systems can be reduced into several different echelon forms
Reduced echelon form
A system is only row equivalent to one REF
Forward phase
Reducing an augmented matrix A into an echelon form
Backward phase
Reducing an augmented matrix A into a reduced echelon form
Basic variables
Variables in pivot columns.
Free variables
Variables that are not in pivot columns. When a system has a free variable the system is consistent but not unique

Subspaces of R^n

A subspace of Rn is any set H in Rn that has three proper­ties:
- The zero vector is in H
- For each u and v in H, the sum u + v is in H
- For each u in H and each scalar c, the vector cu is in H
Zero subspace is the set containing only the zero vector in Rn
Column space is the set of all linear combin­ations of the columns of A.
Null space (Nul A) is the set of all solutions of the equation Ax = 0
Basis for a subspace H is the set of linearly indepe­ndent vectors that span H
In general, the pivot columns of A form a basis for col A
The number of vectors in any basis is unique. We call this number dimension
The rank of a matrix 𝐴, denoted by rank 𝐴, is the dimension of the column space of 𝐴
Determine whether b is in the col A. Solution: b is only in col A if the equation Ax = b has a solution

Algebraic properties of a vector

u + v = v + u
(u + v) + w = u + (v + w)
u + (-u) = -u + u
c(u + v) = cu + cv
c(du) = (cd)u