Exam (elaborations)
Linear Algebra Summary
- Course
- Linear Algebra III (MAT3701)
- Institution
- University Of South Africa
Linear Algebra Summary
[Show more]
Content preview
lOMoARcPSD|6392334Linear Algebra Summary
P. Dewilde K. Diepold
October 21, 2022
Contents
1 Preliminaries .......................................................................................................................................... 2
1.1 Vector Spaces ......................................................................................................................................... 2
1.2 Bases ...................................................................................................................................................... 4
1.3 Matrices ................................................................................................................................................. 5
1.4 Linear maps represented as matrices ................................................................................................. 6
1.5 Norms on vector spaces...................................................................................................................... 11
1.6 Inner products ..................................................................................................................................... 13
1.7 Definite matrices ................................................................................................................................. 14
1.8 Norms for linear maps ........................................................................................................................ 14
1.9 Linear maps on an inner product space ............................................................................................ 14
1.10 Unitary (Orthogonal) maps .............................................................................................................. 15
1.11 Norms for matrices ........................................................................................................................... 15
1.12 Kernels and Ranges .......................................................................................................................... 16
1.13 Orthogonality .................................................................................................................................... 17
1.14 Projections......................................................................................................................................... 17
1.15 Eigenvalues and Eigenspaces ........................................................................................................... 18
2 Systems of Equations, QR algorithm ................................................................................................ 19
2.1 Jacobi Transformations ...................................................................................................................... 19
For more information. Email: morrisprofessionals@gmail.com
, lOMoARcPSD|6392334
2.2 Householder Reflection ...................................................................................................................... 20
2.3 QR Factorization .................................................................................................................................. 20
2.3.1 Elimination Scheme based on Jacobi Transformations................................................................. 20
2.3.2 Elimination Scheme based on Householder Reflections............................................................... 21
2.4 Solving the system Tx = b.................................................................................................................... 22
2.5 Least squares solutions ...................................................................................................................... 22
2.6 Application: adaptive QR .................................................................................................................... 24
2.7 Recursive (adaptive) computation .................................................................................................... 25
2.8 Reverse QR........................................................................................................................................... 27
2.9 Francis’ QR algorithm to compute the Schur eigenvalue form ........................................................ 28
3 The singular value decomposition - SVD......................................................................................... 30
3.1 Construction of the SVD...................................................................................................................... 30
3.2 Singular Value Decomposition: proof ................................................................................................ 31
3.3 Properties of the SVD .......................................................................................................................... 32
3.4 SVD and noise: estimation of signal spaces ...................................................................................... 34
3.5 Angles between subspaces ................................................................................................................. 37
3.6 Total Least Square - TLS ..................................................................................................................... 37
1 Preliminaries
In this section we review our basic algebraic concepts and notation, in order to establish a common
vocabulary and harmonize our ways of thinking. For more information on specifics, look up a basic
textbook in linear algebra [1].
1.1 Vector Spaces
A vector space X over R or over C as ’base spaces’ is a set of elements called ’vectors on which
’addition’ is defined with its normal properties (the inverse exists as well as a neutral element called
zero), and on which also ’multiplication with a scalar’ (element of the base space is defined as well,
with a slew of additional properties.
For more information. Email: morrisprofessionals@gmail.com
, lOMoARcPSD|6392334
Concreteexamplesarecommon:
z
1
-3
5
in R 3 : −3
1 5 x
y
5+ j
in C 3: − 3 − 6j ≈ R6
2+2 j
The addition of vectors belonging to the same Cn (Rn) space is defined as:
and the scalar multiplication: a ∈ R or a ∈ C:
Example
The most interesting case for our purposes is where a vector is actually a discrete time sequence {x(k)
: k = 1···N}. The space that surrounds us and in which electromagnetic waves propagate is mostly
linear. Signals reaching an antenna are added to each other.
Composition Rules:
The following (logical) consistency rules must hold as well: x
+ y = y + x commutativity
(x + y) + z = x + (y + z) associativity
0 neutral element x + (−x)
= 0 inverse
distributivity of ∗ w.r. + consistencies
Vector space of functions
For more information. Email: morrisprofessionals@gmail.com
, lOMoARcPSD|6392334
Let X be a set and Y a vectorspace and consider the set of functions
X → Y.
We can define a new vectorspace on this set derived from the vectorspace structure of Y:
(f1 + f2)(x) = f1(x) + f2(x)
(af)(x) = af(x).
Examples:
(1)
f1 f2
+ = f1 + f2
(2)
+ =
[x 1 x 2 ··· x n ]+[ y 1 y 2 ··· y n ]=[ x 1 + y 1 x 2 + y 2 ··· x n + y n ]
As already mentioned, most vectors we consider can indeed be interpreted either as continous time
or discrete time signals.
Linear maps
Assume now that both X and Y are vector spaces, then we can give a meaning to the notion ’linear
map’ as one that preserves the structure of vector space:
f(x1 + x2) = f(x1) + f(x2) f(ax) =
af(x)
we say that f defines a ’homomorphism of vector spaces’.
1.2 Bases
We say that a set of vectors {ek} in a vectorspace form a basis, if all its vectors can be expressed as a
unique linear combination of its elements. It turns out that a basis always exists, and that all the bases
of a given vector space have exactly the same number of elements. In Rn or Cn the natural basis is given
by the elements
0
For more information. Email: morrisprofessionals@gmail.com
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller morrisacademiamorrisa. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R46,57. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
64438 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now