"Eigenvectors... change at most by a scalar factor when that linear transformation is applied."
"Linear algebra is the backbone of calculations behind neural network algorithms."
"Vectors with that property have a special name. They're called eigenvectors."
"The Graham-Schmidt formula removes components in previous directions to achieve orthogonality."
"But in this video I hope to show you something that you're going to do in linear algebra that you've never done before, and that it would have been very hard to do had you not been exposed to these videos."
"The determinant of an upper triangular matrix is the product of the diagonal entries."
"In Quantum Computing, we use bracket notation, which is really just a way to disguise linear algebra."
"So in other words, the trace of A equals the trace of QAQ^(-1)."
"Linear algebra is unreasonably effective at studying the rest of mathematics."
"Translate that mathematical structure into the language of linear algebra."
"Studying inside of this linear algebra framework you get more deep understanding of the original mathematical structure."
"So the multiplying-by-constant function is a linear map corresponding to a matrix of this form."
"You can create any arrow of any length pointing in any angle through linear combinations of basis vectors."
"There always exists U and V such that A can be decomposed as U * Sigma * V transpose."
"Singular value decomposition works for all general n by n matrices."
"This process of finding a singular value decomposition is quite simple."
"We're going to use this same idea that comes right out of linear algebra and come to solve lu equals to f using this eigen decomposition technique."
"Self-adjoint operators are very important."
"Numpy supports n dimensional arrays, provides numerical Computing tools useful for linear algebra and forer transform."
"So when you manipulate a vector with a matrix, what you're actually doing is nothing more than rotating and stretching. That's it."
"This is where we show you how it relates to eigenvalues and eigenvectors."
"So, take each one of these, interchange rows and columns, and complex conjugate, and that's the Hermitian conjugate operator."
"...whenever I get into the linear algebra, I'm actually going to get a lot into the math and explaining what the functions actually do..."
"...there's also another version...if you do not need them, you can just type in eigenvalues like that and then you won't get them..."
"All right, so there is a rough overview of a lot you can do in regards to linear algebra inside of numpy."
"Enjoy scripted this in in Blender with those three vectors you know the first vector becomes the first column of the matrix the second vector becomes the second column of the matrix."
"The activation function is essential because if you didn't have your activation function, this sequence of linear transformations is equivalent to just a single linear transformation."
"Positive definite matrices come from least squares and all sorts of physical problems."
"If a matrix is positive definite, then its inverse is also positive definite."
"Every matrix is similar to its Jordan form."
"Circulant matrices are connected with the discrete Fourier transform."
"The linear algebra for Fourier matrices is incredibly important."
"The linear algebra is the secret to everything."
"The projection of vector X onto vector V is given by X dot V over magnitude of V times u."
"By linear algebra we know that the determinant of a product AB is equal to the determinant of A times the determinant of B."
"I can solve AX = B exactly when B is in the column space."
"The identity matrix in R tells you which columns at the start were independent."
"Every matrix defines a linear transformation on the plane."
"Differential calculus is all linear algebra."
"The rows are the outputs. The columns are the inputs, because it has to be this linear operator."
"It's the linear operator that takes in a vector and gives you a scalar."
"We have transformed our three by three matrix into its diagonalized representative."
"When the matrix acts on this vector, the vector stays the same; it's just multiplied by a value."
"The spectral theorem: a symmetric matrix can be factored into an orthogonal matrix times diagonal times the transpose of that orthogonal matrix."
"Every row will have a one somewhere, and that means you cannot have an inconsistent equation."
"If you want to know the subject of today's class, it's Ax = B."
"Using the pseudo-inverse... deals with zero eigenvalues and zero singular values by saying their inverse is also 0, which is kind of wild."
"The condition number is a ratio of the largest to the smallest singular value."
"We're taking a linear combination of V and A in a non-trivial way."
"One really important thing in linear algebra is how we manipulate vectors, how we scale them, and how we combine them."
"Generalized cross-validation is a superfast linear algebra approximation to leave-one-out cross-validation."
"This is really about the rank of a matrix, what it means, how do we get it kind of brute force, and why is it useful for us in data science at all."
"To be linear, a transformation has to preserve vector addition and scalar multiplication."
"There's more to Quantum Mechanics than just dealing with the wave function; we can do some interesting things with the linear algebra structure."
"The whole game of a lot of this linear algebra is not can I get a solution, it's always about how fast can you do it."
"If you have a product of two matrices, they both get a rank one update, so the update you make to each of them in aggregate is richer."
"PCA is a linear transformation which transforms the given data from n-dimensional space to another space with the same number of dimensions."
"Eigenvalues and eigenvectors are incredibly powerful tools."
"The determinant of A minus lambda I is equal to zero."
"Eigenvalues and eigenvectors are very important mathematical machinery that we're going to use over and over again in future lectures."
"The answer to this question is hidden in the eigenvalues of A."
"The really important equation is AX equals lambda X."
"This is a very important set of numbers lambda and vectors X."
"It's only really valid to do the matrix multiplication in the condition where the number of columns on the left-hand side is equal to the number of rows on the right-hand side."
"Matrices are, some would argue, the essence of linear algebra and really are incredibly important in a great many fields of science and engineering."
"We say a set spans a vector space if anything in the vector space can be written as a linear combination of the spanning set."
"The linear algebra library in Julia actually makes heavy use of the type system and that improves performance."
"XLA, which stands for Accelerated Linear Algebra, is a domain-specific compiler for linear algebra."
"The determinant function is a function \( D \) which maps from the \( n \times n \) matrices to the real numbers."
"Numpy provides us numerous different math functions, ways of working with linear algebra, and ways of manipulating data."
"Q inverse is the same as Q transpose."
"It's exactly an expansion in an orthonormal basis."
"In essence, random matrix theory is the happy marriage between linear algebra and probability theory."
"Eigenvectors and eigenvalues are one of the most fundamental concepts in linear algebra and are extremely important and widely applicable to a lot of different things."
"They span R3, so they are a basis for R3."
"Linear algebra is really an essential class for pretty much all computer scientists and mathematics majors."
"This is where linear algebra and differential equations meet, and it is wonderful."
"Eigenvectors are orthogonal; they span a vector space; they represent principle directions."
"The real picture is Y equals AX plus V."
"Subspaces always pass through the origin by definition."
"The existence of bases allows us to find a subspace that, in some sense, lifts the quotient space."
"You can always take a linearly independent set and extend it to a basis."
"Every vector W in V is uniquely expressed as a linear combination."
"The beautiful thing about a basis is that every combination is unique."
"The matrix that helps us do this is called the Jacobian matrix."
"The Jacobian and the inverse Jacobian matrices are indeed inverses."
"The Gram-Schmidt process is a procedure that you can follow to obtain an orthonormal set of vectors within an inner product space."
"The identity matrix is a square matrix where all of the diagonal entries are ones and all of the other entries in the matrix are zeros."
"The problem based on linear algebra is not solvable because the number of unknowns is more."
"The amazing thing is that L of C1 U1 plus C2 U2 is also 0."
"One of the coolest properties in linear algebra is that when you use the eigenvalues and eigenvectors of A to diagonalize the system, then it gives you a really easy way of computing the matrix exponential."
"A matrix can be decomposed in many ways, one of which is the singular value decomposition."
"Linear dependence implies redundancy."
"Every vector in the given vector space can be written uniquely as a linear combination of those basis elements."
"What I'm going to talk about today will be more about the linear algebra that's behind all of quantum mechanics."
"The rank nullity theorem follows; we have got a proof of one of the most important theorems in all of linear algebra."
"Linear functions have a constant rate of change."
"The determinant of three vectors gives me the volume of a little parallel pipette, a little box with edges u, v, and w."
"The direction of projection is a vector W."
"The generator matrix provides a concise and efficient way to represent a linear block code."
"With linear codes, the sum of any two valid codewords is another valid codeword."
"It's so useful to understand linear algebra because it helps us use computers to solve our problems."
"Follow me on this journey into the next lesson and beyond; we will be studying linear algebra step by step."
"The four big problems of linear algebra turn out to a good way to describe the answer is as a factorization of the matrix."
"These have terrific eigenvalues and especially eigenvectors."
"The exact solution would come from the eigenvalues and eigenvectors."
"Linear algebra is fundamentally not about matrices."
"Any function of the matrix, we could define the exponential of a matrix."
"If all the eigenvalues Lambda of a are real and distinct, then essentially my eigenvectors T span RN."
"We're getting very close to being done with E to the A, because this is the last piece of the puzzle."
"This A matrix is easy in some sense because I have two orthogonal eigen vectors."
"Any non-singular matrix you can factor into a permutation, a lower triangular, and an upper triangular matrix."
"Basis vectors and vector components are like two sides of the same coin; we can't talk about one without also talking about the other."
"This Matrix exponential E to the A T is going to be a matrix."
"If I know the action of L on a finite set of things, namely the basis vectors, then I actually know the action of L anywhere."
"A linear map is really determined by its action on the basis vectors for the space."
"This is a way to kind of patch up a whole linear algebra; linear algebra doesn't know how to translate because it works on vectors and not points."
"Now remember that if I take two points and I subtract them, I get a vector."
"By adding a fourth coordinate where the coordinate is one if I'm a point and zero if I'm a vector, allows me to have a consistent picture of linear algebra that deals with both points and vectors at the same time."
"The basic object that we deal with in linear algebra is something called a vector space, equipped with an addition and a scalar multiplication operation."
"Linear algebra has gradually taken over a much bigger part of today's tools."
"The product rule just works for matrices and vectors as well as for scalars."
"The transformations of the coefficients with the basis vectors exactly compensate."
"The image is the span of the columns of the matrix."
"If you do want to be a mathematics major in college, these are the foundations of linear algebra."
"I can already do GCD, I can already get my null basis vectors, now I can do all the things."
"The best \(X\) is the eigenvector, and the ratio is the eigenvalue."
"Many courses on linear algebra never reach this key idea of positive definiteness that ties it all together."
"What's nice about that left side, what fact am I going to use about Q? Q transpose Q is the identity."
"You asked for some linear algebra, and you got it."
"A is the matrix that takes differences of the U's, and then A transpose A takes second differences."
"Everything you know about a matrix shows up somehow in its eigen vectors and eigen values."
"Linear independence is a very fundamental concept."
"An eigen vector is any nonzero vector which has its direction preserved under the mapping F."
"Linear problems have the add-add property."
"The corresponding y is generated by taking the dot product of theta and the features of the input and adding a random Gaussian noise."
"The columns of T in T inverse A T equals J are called generalized eigenvectors."
"When you see X dot equals AX and all of the eigenvalues are lambda, what the Jordan blocks in A mean is that if A is diagonalizable, the only thing you will see in the solution will be exponentials."
"The exponential of a matrix is a linear combination of I, A, A squared, up to A to the n minus one."
"The awesome thing is you're actually able to do exactly the same thing using matrices."
"Assume \( A \) is diagonalizable, we have \( T^{-1}AT = \Lambda \)."
"Each of you has a piece of electronics with you that involves a non-trivial Jordan block."
"Matrices are an n by n array of special numbers; they have a meaning when they multiply a vector, they do something."
"The word 'positive definite' just brings the whole of linear algebra together."
"Matlab is a tool that's made for linear algebra."
"If we found the eigenvalues and eigenvectors of A, that essentially defines a new coordinate system where it's easier to represent the dynamics and solve them."
"Every vector in the vector space can be written as the linear combination of the basis vectors in one and only one way."
"Inverses are always going to be symmetric in the line Y equals X."
"The solution is very easy to see if it's in reduced row Echelon form."
"The matrix A is called non-singular provided A inverse exists."
"The rank of a matrix is not affected if I multiply it with invertible matrices."
"The spectrum of a matrix and the spectrum of its transpose are the same."
"The eigen vectors of a matrix associated with different eigen values are linearly independent."
"There always exists an orthonormal set of eigen vectors of S which form a basis."
"The norm of a matrix, also called the matrix norm, spectral norm, L2 norm... it's the square root of the largest eigenvalue."
"We will also focus our attention on the solution of matrix equations of the form AX is equal to B."
"We would be determining a scalar lambda and a non-zero vector x such that this matrix equation AX is equal to lambda X is satisfied."
"Linear algebra can be thought of as the study of linear transformations."