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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 GramSchmidt 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: ntuple
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
GaussJordan 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 ntuples,
: set of m × n matrices,
is identified with
basic unit vectors: (1 in jth
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 ith row
and jth column
ijth cofactor of A:
chess board rule :
inductive definition:
cofactor expansion along first row
cofactor expansion:
along ith row
along jth 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 ith column by b
adjoint of A: transpose of matrix of
cofactors
adjoint formula for inverse :
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