Linear Algebra (English)

   Tuesday, July 18, 2023   

Starting a new Lecture Notes Series on Linear Algebra

Youtube Lecture Playlist Credits 

Channel Name: Dr. Mathaholic

So Let Us Start to This Journey of Learning 

Linear Algebra By Lecture Notes together!

Lecture Description of this above Topic:

Lecture 1: Some nice properties that real numbers satisfies but Matrices don't.

Lecture 2: Any square matrix can be written as sum of Symmetric and Skew Symmetric matrix and that too UNIQUELY

Lecture 3: Tricks to find examples on Symmetric & Skew symmetric matrix. Diagonal entries of sk sym matrix = 0.

Lecture 4: Can a non zero matrix be both symmetric as well as skew symmetric?

Lecture 5: What can you say about determinant of a Skew Symmetric matrix of odd order? Is it Zero? Why?

Lecture 6: Can we have a non-zero #matrix which is #diagonal as well as skew-symmetric?

Lecture 7: #shortcut to find #determinant of a #matrix using #properties . #mathematics

Lecture 8: Orthogonal matrix & examples. Inverse, transpose, arithmetic operations between orthogonal matrices.

Lecture 9: Rows & columns of orthogonal matrices are orthogonal. Moreover length of row & column vectors are 1.

Lecture 10: Any 2 by 2 orthogonal matrix is either a rotation matrix or a reflection matrix.

Lecture 11: Row / Column operations on a matrix is equivalent to Pre / Post Matrix Multiplication.

Lecture 12: Example when (ab)^inverse is not equal to a^inverse*b^inverse

Lecture 13: Linear Algebra Question on row reduced echelon form in GATE 2021

Lecture 14: How Linear Combination concept evolved from your high school mathematics to Vectors spaces!

Lecture 15: Solution to traffic problem using LINEAR ALGEBRA.

Lecture 16: Linearly Independent and dependent vectors:: concept and examples

Lecture 17: Subsets and Supersets of Linearly Independent and Linearly dependent sets.

Lecture 18: Rank of a Matrix: Maximum number of linearly independent row or column vectors

Lecture 19: Existence of solution to System of Linear equations.

Lecture 20: Matrix rank example (CSIR NET 2019 question - Linear algebra)

Lecture 21: VECTOR SPACE: Looking at the Properties and an Example simultaneously.

Lecture 22: Matrices and Vector Space Structure. Examples and Non-Examples.

Lecture 23: Is R^2 always a vector space over R? One has to be careful before answering Yes

Lecture 24: Tricks to determine whether a given subset of R^n is a subspace or not.

Lecture 25: If {0} is the only linearly dependent set of V then the dimension of V is???

Lecture 26: Example of a subset of R^2 which is not a subspace??

Lecture 27: Number of elements in a Subspace are infinite!!

Lecture 28: Examples on Subspaces and Non Subspaces of Matrix Space!

Lecture 29: Examples of SubSpaces and Non SubSpaces of Polynomial Space

Lecture 30: Is Addition, Scalar Multiplication, Intersection and Union of Subspaces is again a Subspace?

Lecture 31: When is Union of Subspaces is again a subspace? Proof and Counterexample.

Lecture 32: Spanning set of subset in Vector Space along with important results.

Lecture 33: Basis and Dimension of a Vector space. How can one construct infinitely many basis subsets?!

Lecture 34: Basis and Dimension for Symmetric Matrices of Order n. Precise as well as Shortcut proof!

Lecture 35: Basis and Dimension for Skew Symmetric Matrices. Precise as well as Shortcut solution.

Lecture 36: Basis and Dimension of Trace 0 and Other such type of Matrices.

Lecture 37: An Example on Basis and Dimension of U, V, U intersection V and U + V of a polynomial space

Lecture 38: Rank Nullity theorem for matrices.

Lecture 39: Linear Algebra question on dimension of subspaces in GATE 2022 Exam

Lecture 40: Coordinate vector with respect to given ordered basis. Is this a well defined concept??

Lecture 41: Motivation to study Change of Basis Matrix along with an example and homework problem

Lecture 42: What the word linear means in Linear Transformations. Concept, Examples and some nice Hints

Lecture 43: What is and Why do we study Kernel of a linear transformation?

Lecture 44: Examples on Image space of a linear transformation

Lecture 45: Linear transformation is one-one function if and only if Kernel contains only singleton zero vector

Lecture 46: Linear map is one one if and only if its onto if and only if its invertible iff bijective

Lecture 47: Steps on how to find a Matrix corresponding to a Linear Transformation.

Lecture 48: What are Eigenvalues and Eigenvectors? Concept and examples using linear transformations.

Lecture 49: Steps to find eigenvalues and eigenvectors along with examples.

Lecture 50: Eigenvalues and Eigenvectors for power of a Matrix?

Lecture 51: Some Nice properties of Eigenvalues.

Lecture 52: Determinant of a matrix is equal to product of eigenvalues. Proof and simple example.

Lecture 53: Properties of Eigenvalues for scalar multiplication and scalar addition of matrices.

Lecture 54: Does a #matrix always have #real #eigenvalues ?

Lecture 55: Can we have an #eigenvector corresponding to two different #eigenvalues ?

Lecture 56: Example of an integral operator which has NO Eigenvalues!!

Lecture 57: Eigenvalues and Eigenvectors of a polynomial

Lecture 58: Relation between Eigenvalues and Eigenvectors of inverse of A and adjoint of A.

Lecture 59: Linear algebra Question on Trace and Determinant of a matrix in GATE 2022 Examination

Lecture 60: Algebraic multiplicity, Eigenspaces and Geometric multiplicity. Any connection between them?

Lecture 61: Proof of - Geometric Multiplicity is less equal Algebraic Multiplicity

Lecture 62: Similar matrices and the ten nice properties that they preserve.

Lecture 63: What is, and importance of, diagonalizable matrices. Examples and uniqueness using examples.

Lecture 64: Given eigenvalues and eigenvectors, how to find a matrix?

Lecture 65: Cayley Hamilton Theorem: Its meaning and application in solving problems

Lecture 66: Linear algebra question on Cayley-Hamilton Theorem in GATE 2022

Lecture 67: Why do we need Inner Product? Motivation to study this topic!

Lecture 68: Infinitely many examples and non-examples of Inner products on vector spaces.

Lecture 69: Infinite examples on Inner Product on Polynomial Vector Space.

Lecture 70: Linear Algebra Question on Non-Singularity and Nullity in GATE 2021 exam

Lecture 71: Example of an inner product on Matrices which helps us to find angle and length.

Lecture 72: Orthogonal Vectors are Linearly Independent vectors but not conversely. Counterexample as well.

Lecture 73: Orthogonal Matrices may not be Orthogonal!!

Lecture 74: Orthogonal and Orthonormal vectors and its connection with Linearly independent vectors!!

Lecture 75: Motivation to study Gram- Schmidt orthogonalization process!

Lecture 76: Gram Schmidt Orthogonalization process (Geometrically). Formula and examples.

Lecture 77: Generalization of Generalized Pythagoras Theorem. (Pythagoras identity in Inner Product space)

Lecture 78: Orthogonal Complement. Concept, Examples, Nice results and Some Tricks.