Linear Algebra Cheat Sheet

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A cheat sheet for linear algebra midterms

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PRs

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

(A^T)^-1= (A^-1)^T

Linear Transformations

Tranformation/mapping	T(x) from Rⁿ to R^m
Image	For x in Rⁿthe vector T(x) in R^mis 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.

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 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	I^mA = A = AIⁿ
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^-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

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