Cheatography
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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. 
Nontrivial solution 
A nonzero vector that satisfies Ax = 0. Has free variable. 
Inverser of a Matrix
C is invertible if CA = I^{n} and AC = I^{n} 
If A is (2x2) then, A^{1} = 
(A^{1})^{1} = A 
(AB)^{1}= B^{1}A^{1} 
(A^{T})^{1}= (A^{1})^{T} 
Linear Transformations
Tranformation/mapping 
T(x) from R^{n} to R^{m} 
Image 
For x in R^{n}the vector T(x) in R^{m}is 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 R^{n} 
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. 
Onetoone 
If each b in the codomain is only the image at most one x in the domain. Solution Uniqueness. 

T is onetoone 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 onetoone. 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 onetoone 
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 onetoone. 
The equation 𝐴𝐱 = 𝐛 has at least one solution for each 𝐛 in R^{n} 
The columns of 𝐴 span R^{n} 
The linear transformation 𝐱 ↦ 𝐴𝐱 maps R^{n} onto R^{n} 
There is an 𝑛×𝑛 matrix 𝐶 such that 𝐶𝐴 = 𝐼. 
There is an 𝑛×𝑛 matrix 𝐷 such that 𝐴𝐷 = 𝐼. 
𝐴^{T} 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 I^{n}. 
A = E^{1}I^{n} and A^{1} = EI^{n}= I^{n} 
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 
I^{m}A = A = AI^{n} 
Commute 
If AB = BA then we say that A and B commute with each others 

(A^{T})^{T} = A 

(A + B)^{T} = A^{T} + B^{T} 

(AB)^{T} = B^{T} A^{T} 

For any scalar r, (rA)^{T} = rA^{T} 


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^{1}U = 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 nonzero 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 R^{n} is any set H in R^{n} 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 R^{n} 
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 
