# LINEAR ALGEBRA NOTES

Contents

1. Systems of linear equations

2. Matrices of a system

3. Gauss elimination

4. Matrices

5. Matrix operations

6. Inverse matrix

7. Determinants

8. Vector spaces

9. Linear independence

10. Bases

11. row, column and null spaces

12. Coordinates

13. Linear transformations

14. Eigenvalues and eigenvectors

15. Diagonalization

16. Bilinear functional

17. Inner product

18. Orthogonal bases and Gram-Schmidt algorithm

19. Least square solution and linear regression

1. Systems of linear equations

**linear equation:**

**variables:**

**coefficients:**

**main coefficient: **

** constant term : **b

**linear system: **m equations, n unknowns

**solution:** n-tuple
satisfying all equations

**consistent system**: has a solution

**inconsistent system: **has no solution

**solution set:** set of all solutions

**equivalent systems:** have the same solution set

**elementary operations on equations :** make equivalent systems

(i) multiply an equation by a nonzero constant

(ii) interchange two equations

(iii) add a constant multiple of an equation to another

**elimination:** use elementary operations to eliminate unknowns

**
fact: **a linear system has no solution, exactly one solution or infinitely
many solutions

**parameters:**used to describe infinitely many solutions

**homogeneous system:**constant terms are 0 (consistent)

**trivial solution:**all variables are 0

2. Matrices of a system

**coefficient matrix :**

**constant vector:
unknown vector:
augmented matrix:**

3. Gauss elimination

**elementary row operations:** (ero) correspond to elementary operations on
equations

(i) multiply a row by a nonzero constant

(ii) interchange two rows

(iii) add a multiple of a row to another row

**row equivalent matrices:** one can be gotten from the other by elementary
row operations

**fact:** linear systems with row equivalent augmented matrices have the same
solution set

**echelon matrix:** the number of leading zeros is strictly increasing in
each row until you get all 0 rows

**Gauss elimination: **use elementary row operations to get echelon form

**leading entry:** first nonzero entry in a row

**leading (pivot) column:** column containing a leading entry

**leading variable:** a variable corresponding to a leading column

**free variable: **not leading

**back substitution:** get solution set from echelon form

(i) set free variables equal to parameters

(ii) solve last nonzero equation for leading variable

(iii) substitute into preceding equation

(iv) continue

** reduced echelon matrix:
**

(i) echelon matrix

(ii) every leading entry is 1

(iii) every leading entry is the only nonzero entry in it's column

**Gauss-Jordan elimination:**use elementary row operations to get reduced echelon form

**fact:**every matrix is row equivalent to a unique reduced echelon matrix

**fact:**system with square coefficient matrix A has unique solution i A is row equivalent to I

**fact:**system with more unknowns than equations is inconsistent or has infinitely many solutions

4. Matrices

**matrix:**rectangular array of numbers

**notation:**

**scalar:**real number

**size of a matrix:**size (A) = m × n if m rows and n columns

**square matrix:**m = n

**diagonal matrix:**

**zero matrix :**O all entries are 0

**identity matrix:**

**(column) vector:**has size n× 1

**row vector:**has size 1 × n

slightly abusive

set of n-tuples,

: set of m × n matrices, is identified with

**basic unit vectors:**(1 in j-th position), column vectors of

**column vectors:**

5. Matrix operations

**matrix addition:** if A, B have
the same size

**matrix subtraction: **

**scalar multiplication: **

** negative matrix :** -A = (-1)A

**properties:
**

A + B = B + A commutative

A + (B + C) = (A + B) + C associative

c(A + B) = cA + cB distributive

(c + d)A = cA + dA distributive

(cd)A = c(dA) associative

**matrix multiplication:**C = AB, size(C) = m × n, size(A) = m × p, size(B) = p× n

**properties:**

A(BC) = (AB)C associative

A(B + C) = AB + AC distributive

(A + B)C = AC + BC distributive

c(AB) = (cA)B = A(cB)

**warning:**

AB ≠ BA in general

**transpose:**where

**properties:**

**fact:**product of diagonal matrices is diagonal

**matrix form of linear system:**

linear combination: of objects is a finite sum of scalar multiples of the objects

**span:**of objects is the set of linear combinations of the objects

**fact:**solution set of homogeneous system is the span of particular solutions (one for each parameter)

6. Inverse matrix

**A invertible**: such that AB = BA = I

**B**is the

**inverse**of A (A is also the inverse of B)

**properties:**

invertible square

inverse is unique if exists, notation

if A is invertible then Ax = b has unique solution

fact: is invertible iff

**elementary matrix:**single elementary row operation

**properties:
**

implies equivalently

inverse ero

**fact:**A invertible i A row equivalent to I

**fact:**A, B row equivalent iff , for elementary

**algorithm for**A

^{-1}:

more generally

7. Determinants

**notation:**

**notation:**after deletion of i-th row and j-th column

**ij-th cofactor of A:**

**chess board rule:**

**inductive definition: **

cofactor expansion along first row

**cofactor expansion:**

along i-th row

along j-th column

**elementary row operations: **

**properties:
**

A triangular implies

detI = 1

implies detA = 0

A invertible iff detA ≠ 0

**Cramer's rule:**detA ≠ 0 implies solution of Ax = b is

where comes from A after replacing i-th column by b

**adjoint of A:**transpose of matrix of cofactors

**adjoint formula for inverse:**

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