Math

1. Experiential Learning
Phase 1
MATHEMATICS
LINEAR ALGEBRA AND IT’S APPLICATIONS
RV College of
Engineering
Go, change the world
Admission No. Name Email Id
1 RVCE22BIS020 PRACHI N prachin.is22@rvce.edu.in
2 RVCE22BCD023 MULA SOHAN mulasohan.cd22@rvce.edu.in
3 RVCE22BCS121 SUHAS GOWDA L suhasgowdal.cs22@rvce.edu.in
4 RVCE22BAI011 NIDHI BC nidhibc.ai22@rvce.edu.in

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PRESENTATION OUTLINE
TABLE OF CONTENTS
INTRODUCTION
THEORY AND VISUALIZATION
COMPUTATIONAL CHEMISTRY
QUANTUM COMPUTING
DIMENSIONALITY REDUCTION
 REFERENCES

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INTRODUCTION
Linear algebra is the study of linear combinations. It is the study of vector spaces, lines and
planes, and some mappings that are required to perform the linear transformations. It
includes vectors, matrices and linear functions. It is the study of linear sets of equations and
its transformation properties.
It is a key concept for almost all areas of mathematics. Linear algebra is considered a basic
concept in the modern presentation of geometry. It is mostly used in Physics and Engineering
as it helps to define the basic objects such as planes, lines and rotations of the object. It
allows us to model many natural phenomena, and also it has a computing efficiency.

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Transformation refers to use movement. We imagine
input vector moving towards the output vector.
A transformation is linear if:
• Origin is fixed.
• All lines (including the imaginary diagonal line)
remain line, without getting curved.
Vector v = xi + yj when linearly transformed:
v = x(transformed i) + y( transformed j)
v = x a + y b
c d
a b
where i lands c d where j lands
MATRIX AS LINEAR TRANSFORMATION

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INVERSE OF A MATRIX RANK
900
𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑟𝑜𝑡𝑎𝑡𝑒 → 900
𝑎𝑛𝑡𝑖𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒
𝐿𝑒𝑓𝑡𝑤𝑎𝑟𝑑 𝑠ℎ𝑒𝑎𝑟 → 𝑟𝑖𝑔ℎ𝑡𝑤𝑎𝑟𝑑 𝑠ℎ𝑒𝑎𝑟 𝑠𝑎𝑚𝑒 𝑎𝑚𝑜𝑢𝑛𝑡
𝐴. 𝐴−1
= 1
Rank =no. of dimensions in a column
space
Null space (kernel): space of all vectors that
become null ( they land on zero vector)

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Factor by which linear transformation changes any area.
NON SQUARE MATRICES
Determinant

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(p*,q*)
Change of Basis Vector

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Basic equation : Ax = λx
where λ : eigen value of A
x: eigen vector ( non zero vector)
Characteristic equation:
|A- λI | = 0
Properties of Eigenvalues
• Eigenvectors with distinct eigen values
are linearly independent.
• Singular matrix: λ=0
• If A is square matrix
λ=0 is not an eigen value of A.
Scalar multiple: a λ → a A
Power: λ𝑛
→ 𝐴𝑛
If p(x) is polynomial of variable x ,
then
p(λ) → p(A)
Inverse: λ−1
→ 𝐴−1
Transpose: λ = 𝐴𝑇
EIGEN VALUES AND EIGEN VECTORS

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APPLICATIONS IN COMPUTATIONAL CHEMISTRY
The Adjacency Matrix
The vertex-adjacency matrix A(G) of a labeled connected graph G with N vertices is the square N x N symmetric
matrix which contains information about the internal connectivity of vertices in G. It is defined as,
Therefore, a nonzero entry appears in A(G) only if an edge connects vertices i and j. For example, the
following vertex-adjacency matrix can be constructed for a labeled graph G
𝐴𝑇 (𝐺) = 𝐴 (𝐺)

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APPLICATIONS IN COMPUTATIONAL CHEMISTRY
Edge-Adjacency Matrix
The edge-adjacency matrix of a graph G, EA = EA(G), is determined by the adjacencies of edges in
G. It is very rarely used. The edge-adjacency matrix is defined as
For example, the following edge-adjacency matrix can be constructed for a labeled graph G
Pair of non-isomorphic graphs which possess the identical edge adjacency matrix have different
vertex-adjacency matrices.

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APPLICATIONS IN COMPUTATIONAL CHEMISTRY
Distance Matrix
The distance matrix D = D(G) of a labeled connected graph G is a real symmetric N x N matrix whose
elements (𝐷)𝑖𝑗are defined as follows:
where 𝑒𝑖𝑗 , is the length of the shortest path (i.e., the minimum number of edges) between the vertices 𝑣𝑖 ,
and 𝑣𝑗 . The length is also called the distance between the vertices vi, and vj thence the term distance matrix.
For example, the following distance matrix can be constructed for a labeled graph G
The distance matrix has found a widespread application in chemistry in both explicit and implicit
forms. The first explicit use of the distance matrix was employed the for studying the permutational
isomers of stereo chemically nonrigid molecules. The distance matrix in explicit form is also used to
generate the distance polynomial and the distance spectrum.

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APPLICATIONS IN COMPUTATIONAL CHEMISTRY
Wiener Index
It is defined at the sum of distances between any two carbon atoms (pairs of nodes) in the molecule.
Mathematically it is represented as:
Where G represents the total atoms in the molecule, u and v are individual carbon atoms and
d(u , v) is the distance in bonds between any two carbon atoms in the shortest path between
any two atoms.
To calculate the Wiener index for a molecule, for each pair of atoms in the structure, count the distance
between atoms. Take the sum of all distances and divide by two. For example in the case of pentane,
which has five nodes:
A
B
C
D
E

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APPLICATIONS IN COMPUTATIONAL CHEMISTRY
The characteristic (spectral) polynomial P(G ; x) of a graph G is the characteristic polynomial of its adjacency matrix,
P(G ; x) = det |x I - A|
where A and I are, respectively, the adjacency matrix of a graph G with N vertices and the N x N unit matrix. A graph
eigenvalue xi is a zero of the characteristic polynomials.
P(G ; xi) = 0 for i = 1, 2 . , N.
The Largest Eigenvalue

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APPLICATIONS IN QUANTUM COMPUTING
Quantum gates: In quantum computing, gates are the building blocks of quantum circuits, and they
operate on quantum bits (qubits). The state of a single qubit is a superposition of the basis states |0⟩
and |1⟩, which can be represented as a column vector:
|ψ⟩ = α|0⟩ + β|1⟩, where α,β𝜖∁
where α and β are complex numbers that satisfy the normalization condition |α|^2 + |β|^2 = 1.
Quantum gates act on these qubit states by applying unitary matrices, which preserve the normalization
and the overall phase of the state.

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APPLICATIONS IN QUANTUM COMPUTING

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DIMENSIONALITY REDUCTION
Dimensionality reduction is the process of reducing the number of features in a dataset while retaining as
much information as possible.
• PCA is a linear algebra technique that transforms high-dimensional data into a lower-dimensional space
by projecting the data onto a new set of orthogonal axes, known as principal components. These
principal components are linear combinations of the original features and are chosen to maximize the
variance of the data in the lower-dimensional space.
• SVD is a matrix factorization technique that decomposes a matrix into three matrices: U, Σ, and V. The Σ
matrix is a diagonal matrix that contains the singular values of the original matrix, and U and V are
orthogonal matrices. SVD can be used for dimensionality reduction by selecting only the top k singular
values and corresponding rows in U and columns in V to obtain a lower-dimensional approximation of the
original matrix.

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REFERENCES
• THEORY AND VISUALIZATION: 3Blue1Brown Youtube Playlist, Geogebra tool for
graphical representation of matrices
• COMPUTATIONAL CHEMISTRY: ‘Computational Chemistry: A Practical guide for
applying techniques to real world problems by Darid C. Young
• QUANTUM COMPUTING: Qiskit website for quantum gates, Quirk Circuit tool
• DIMENSIONALITY REDUCTION: A Practical Introduction to PCA In Python- Deakos
Data Science, Geeks for Geeks for Singular Value Decomposition

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Thank You