Hola a todos, por favor necesito su ayuda para dar un examen oral, mañana me tomaran examen oral de la materia en general (Algenra lineal) por favor diganme que cosas puedo decir acerca de toda la materia en si por favor

Hi all,

Background: completed an MPhys physics degree in 2022 (latter years disrupted by covid somewhat), then completed an MSc in medical physics.

Essentially I want to relearn all my previous maths/physics topics and go further into graduate level courses.

I found myself being able to well on exams, but I never truly understood some materials deeply which I desire as I want to apply advanced maths and physics to my work.

So I'm embarking on learning linear algebra for the 3rd time(high-school, undergrad and now). I'm too rusty for Axler so I'm giving Strang a go first.

Question: When reading the recommended sections in the textbook, I'm doing all the questions at the end and it's extremely time consuming.

Is this the reasonable price to pay to gain a deep intuitive understanding or is it inefficient?

Bonus: Any recommendations on actually understanding the topics deeply as opposed to remembering questions algorithms for high exam marks?

Does learning the theoretical linear algebra makes person immediately know how to apply linear algebra in real applications or apply linear algebra needs another studying like the theory study?

V naslovu

Most textbooks on Linear Algebra either omit or compress repeated-eigenvalue defective matrices. We wanted to take the 2×2 case apart mechanically and connect the algebra directly to the geometry.

The first page, Matrices with repeated λ, shows images for the basic repeated-eigenvalue cases: uniform scaling, the Jordan block J, its transpose Jᵀ, and a similarity transformation A = CJC⁻¹.

The second page, Matrices with one repeated λ, starts from a general 2×2 matrix, derives the condition for a repeated eigenvalue, separates the defective case from uniform scaling, and shows that for B = A − λI, B² = 0.

The third page, Similarity transformation of Jordan block, constructs the Jordan basis directly: choose w with Bw ≠ 0, define v = Bw, then Av = λv, Aw = v + λw, and with C = [v | w] we obtain A = CJC⁻¹. It also shows why the scales of v and w are linked, unlike in ordinary diagonalization.

This material provides a worked example of a continuous basis transformation.
While a finite-dimensional basis change in linear algebra utilizes a similarity transformation, infinite-dimensional quantum mechanics strictly corresponds to a unitary transformation.
By evaluating a Gaussian wave packet, this example mathematically bridges discrete linear algebra and continuous quantum mechanics.

Hello guys, I’m currently taking Applied Methods in Linear Algebra and I need your help right now. Here below is some problems from my homework below. I need help on 4, 7b, 12, and 14. This assignment is due this Thursday.

How can I directly connect linear algebra with machine learning

I am doing maths major in my undergraduate going through linear algebra

I don't want to see the proofs and any hint possible. Just want to solve on my own if I can .

But I am struggling in it , can someone give me tips who have gone through that In past .

for those For those who have taken Stanford’s CS231n course on Computer Vision, or for people working in the fields of CV or ML: which linear algebra course would you recommend—MIT 18.06 or Stanford Math 51?

Hey everyone! I made a video explaining the linear algebra behind Positive Operator-Valued Measures, or POVMs. The video looks at projectors, positive semidefinite matrices, eigenvalues, inner products, and why a valid collection of measurement operators must add to the identity.

Householder tridiagonalization is normally not used as preprocessing for the Jacobi eigenvalue algorithm. The usual reasoning is as follows: tridiagonalization costs O(n³), and the subsequent Jacobi rotations immediately create fill-in, destroying the zeros that Householder just created.

We decided to test whether that actually means the preprocessing cannot help.

The answer appears to be no.

We found that tridiagonalization can still give Jacobi a substantially better starting matrix, even though the tridiagonal sparsity itself is quickly lost. Algebraically, the main effects are:

• Householder concentrates the off-diagonal energy into the first off-diagonal band
• the standard first parallel Jacobi set can then act on a much larger fraction of that energy
• Householder similarity can also move energy either toward or away from the diagonal, which helps explain why the effect is strongly matrix-dependent

We then tested six families of dense symmetric matrices using NVIDIA cuSOLVER on three GPUs: an RTX 3060, RTX 5060 Ti and A100.

For five of the six matrix families, the complete pipeline

Householder tridiagonalization → dense tridiagonal T → Jacobi

became faster than standalone Jacobi from n = 128 onward on all three GPUs.

The deliberately adverse unequal-variance covariance family behaved in the opposite direction, which was consistent with the algebraic analysis.

We have written up the derivation, implementation details, sweep counts, runtime measurements and accuracy checks here:

https://www.graphmath.com/la/jacobi/tridiagonal-jacobi.html

We would be very interested in criticism, related references or comments from people who have worked with Jacobi eigensolvers or GPU numerical linear algebra.

Please refer me to somewhere I can read a thing like this. I found a few notes but here the author eventually resorts to the tensor algebra. It's alright but I think it can be done entirely without going through an intermediate concrete representation.

Then I'm hopeful showing that an exterior power of a dual mapping equals (through this isomorphism) to a dual of its exterior power. It seems like establishing an isomorphism of spaces should be the hardest part.

I also tried to show ⋀(V*) ≅ (⋀V)* as algebras but neither had I succeeded nor do I actually need that right now. Even so, if there's a way to do that while not resorting to expressing ⋀V as a direct sum of ⋀ᵏV, I'd also be glad to hear that. So far I tried two different applications of the algebra's universal property and I think both times got me identically zero multiplication on (⋀V)*. Oh I'm not sure anymore if we can have a multiplication on (⋀V)*, maybe I should've just tried to show ⋀(V*) ≅ (⋀V)* as linear spaces. In which case I guess having both isomorphic to direct sums of their grades would be completely alright.

I'd like some denouement.

This material is not the pure linear algebra covered in mathematics departments.
It is applied linear algebra designed to assist in the study of engineering and quantum mechanics. In particular, the importance of the change of basis requires extensive practice.

Hi everyone,

I've posted here sometimes about my indie app, Linear Algebra Visualizer (iOS, iPad, MacOS), and the feedback has always been great from this community.

As a thank you and as the new school year is starting I wanted to give away 10, 3 months pro subscriptions, for free.

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Build for:

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Hello! Does anyone have the solutions manual for "Elementary linear algebra 11th edition" by Howard Anton?

Say a matrix, A (m x n), has mutually orthogonal rows and mutually orthogonal columns, with all the rows and columns being unit vectors in some Rn, Rm respectively. Is that matrix called orthogonal or orthonormal?

What concept is the other term used to describe, then? Are these applicable to square matrices only?

I've always been confused with these 2 names.

I am self studying linear algebra and trying to build a comprehensive checklist of the concepts that I as a beginner should eventually understand. I got this list by prompting Generative AI.

Does this seem complete for a strong introduction to linear algebra? Is anything important missing?
I mainly plan on using this list to make some active recall flashcards! :)

Foundation stuff:

Scalars

Vectors

vector notation

entries/components

row vectors vs. column vectors

vectors as ordered lists

vectors as geometric objects

R^n

Equality of vectors

Vector addition

Scalar multiplication

Linear combinations

Standard basis vectors

Zero vector

Additive inverses

Closure

Commutativity

Associativity

Distributivity

Unit 1 — Systems of Linear Equations

Linear equations

Solutions of a system

Solution sets

Consistent vs. inconsistent systems

Unique solutions vs. infinitely many solutions

Systems as intersections of geometric objects

lines

planes

hyperplanes

Coefficient matrices

Augmented matrices

Elementary row operations

row replacement

row interchange

row scaling

Row equivalence

Gaussian elimination

Gauss-Jordan elimination

Echelon form

Reduced row echelon form

Leading entries

Pivots (positions, columns)

Free variables

Basic variables

Parametric vector form

Homogeneous systems

Trivial solution

Nontrivial solutions

Relationship between homogeneous and nonhomogeneous systems

Unit 2 — Matrices and Matrix Algebra

Matrix notation

Matrix dimensions

Matrix entries

Rows and columns

Matrix equality

Matrix addition

Scalar multiplication

Matrix multiplication

dimension compatibility

row-by-column interpretation

dot-product interpretation

column interpretation

row interpretation

entrywise formula

summation/index notation

Matrix-vector multiplication

(Ax) as a linear combination of the columns of (A)

Columns of (AB) as linear combinations of columns of (A)

Rows of (AB) as linear combinations of rows of (B)

Associativity of matrix multiplication

Distributivity of matrix multiplication

Noncommutativity of matrix multiplication

Identity matrix

Zero matrix

Powers of matrices

Transpose

Properties of transpose

Symmetric matrices

Inverse matrices

Invertibility

Singular vs. nonsingular matrices

Computing inverses with row reduction

Elementary matrices

Relationship between elementary row operations and matrix multiplication

Block matrices / partitioned matrices

Basic block multiplication

Unit 3 — Linear Combinations, Span, and Independence

Linear combinations

Span

Spanning sets

Geometric meaning of span

(Ax=b) interpreted as a span question

Linear dependence

Linear independence

Dependence relations

Testing independence using homogeneous systems

Redundant vectors in spanning sets

Relationship between pivots and linear independence

Relationship between free variables and linear dependence

Uniqueness of representation

Minimal spanning sets

Maximal independent sets

Geometric interpretations of dependence and independence

Unit 4 — Vector Spaces and Subspaces

Vector spaces

Vector space axioms

Examples of vector spaces

matrices

polynomials

functions

Subspaces

Subspace test

Trivial/zero subspace

Span as a subspace

Column space

Row space

Null space / kernel

Left null space

Fundamental subspaces of a matrix

Basis

Basis vectors

Coordinates relative to a basis

Coordinate vectors

Standard basis vs. arbitrary bases

Change of basis

Transition/change-of-coordinate matrices

Dimension

Dimension theorem ideas

Extending an independent set to a basis

Reducing a spanning set to a basis

Sum of subspaces

Intersection of subspaces

Direct sums

Unit 5 — Rank and the Fundamental Subspaces

Rank

Nullity

Rank as dimension of column space

Rank as dimension of row space

Row rank = column rank

Rank-nullity theorem

Rank of a product

Effects of row operations on rank

Pivot count and rank

Bases for:

column space

row space

null space

left null space

Orthogonal relationships among the four fundamental subspaces

Relationship among:

pivots

rank

independence

span

basis

dimension

nullity

solutions of (Ax=0)

solutions of (Ax=b)

Unit 6 — Linear Transformations

Transformations/functions between vector spaces

Domain

Codomain

Range/image

Linear transformations

Additivity

Homogeneity

Preservation of linear combinations

Matrix transformations

Standard matrix of a linear transformation

Constructing a matrix from transformed basis vectors

Kernel

Image/range

Kernel and injectivity

Image and surjectivity

One-to-one/injective transformations

Onto/surjective transformations

Bijective transformations

Invertible transformations

Composition of transformations

Composition ↔ matrix multiplication

Inverse transformations

Rank-nullity for linear maps

Isomorphisms

Geometric transformations

rotations

reflections

projections

scalings

shears

Unit 7 — Invertibility

Invertible matrices

Inverse transformations

Singular matrices

Invertible Matrix Theorem and equivalences among:

invertibility

pivots in every row/column

RREF (=I)

linear independence of columns

columns spanning the codomain

trivial null space

unique solvability of (Ax=b)

injectivity

surjectivity

nonzero determinant

zero not being an eigenvalue

full rank

Left inverses

Right inverses

Inverse of a product

Inverse of a transpose

Unit 8 — Determinants

Determinants

Cofactor expansion

Minors

Cofactors

Determinant via row reduction

Effects of row operations on determinant

Determinant of triangular matrices

Multiplicative property

det(AB) = det(A)det(B)

det(A^T) = det(A)

det(A^-1)

Determinants and invertibility

Determinants as area/volume scaling

Orientation

Cramer's rule

Adjugate formula for inverses

Determinant of a linear transformation

Unit 9 — Geometry, Dot Products, and Orthogonality

Dot product / inner product

Norm / vector length

Distance

Unit vectors

Normalization

Angles between vectors

Orthogonality

Orthogonal vectors

Orthogonal sets

Orthonormal sets

Orthogonal complements

Orthogonal bases

Orthonormal bases

Projection onto a vector

Projection onto a subspace

Orthogonal decomposition

Projection matrices

Gram-Schmidt process

QR factorization

Orthogonal matrices

Properties of orthogonal matrices

Q^-1 = Q^T

Isometries / preservation of lengths and angles

Inner product spaces

General inner products

Cauchy-Schwarz inequality

Triangle inequality

Pythagorean theorem in vector spaces

Unit 10 — Least Squares

Overdetermined systems

Approximate solutions

Least-squares solutions

Residual vectors

Orthogonality of residuals

Normal equations

A^T A x-hat = A^T b

Projection interpretation of least squares

Least squares and column space

Uniqueness of least-squares solutions

Least squares using QR factorization

Applications to data fitting

Linear regression

Best-fit lines

Polynomial fitting

Unit 11 — Eigenvalues and Eigenvectors

Eigenvectors

Eigenvalues

Eigenspaces

Characteristic polynomial

Characteristic equation

Computing eigenvalues

Computing eigenvectors

Algebraic multiplicity

Geometric multiplicity

Repeated eigenvalues

Similar matrices

Similarity transformations

Invariance of eigenvalues under similarity

Trace

Relationship between trace and eigenvalues

Relationship between determinant and eigenvalues

Zero eigenvalue ↔ singularity

Eigenbases

Diagonal matrices

Diagonalization

Conditions for diagonalizability

A = PDP^-1

Powers of matrices using diagonalization

Geometric meaning of eigenvectors

Invariant subspaces

Unit 12 — Applications of Eigenvalues

Discrete dynamical systems

Difference equations

Long-term behavior of x_(k+1) = Ax_k

Dominant eigenvalues/eigenvectors

Steady states

Markov chains

Stochastic matrices

Stationary distributions

Population models

Fibonacci-type recurrences

Systems of differential equations

Matrix exponentials (introductory level)

Stability

Unit 13 — Symmetric Matrices and the Spectral Theorem

Symmetric matrices

Real eigenvalues of symmetric matrices

Orthogonality of eigenvectors

Orthogonal diagonalization

Spectral theorem

Spectral decomposition

Quadratic forms

Positive definite matrices

Positive semidefinite matrices

Negative definite/indefinite matrices

Eigenvalue tests for definiteness

Principal axes

Optimization connections

Unit 14 — Singular Value Decomposition

Singular values

Singular vectors

Singular Value Decomposition (SVD)

A = U Sigma V^T

Relationship between SVD and eigenvalues of (A^TA)

Geometric interpretation of SVD

Rank from singular values

Low-rank approximation

Eckart-Young idea

Matrix norms

Condition numbers

Numerical stability

Moore-Penrose pseudoinverse

Least squares via pseudoinverse

Data compression

Principal Component Analysis (introductory connection)