What is Linear algebra: Definition and 999 Discussions

Linear algebra is the branch of mathematics concerning linear equations such as:





a

1



x

1


+
⋯
+

a

n



x

n


=
b
,


{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:




(

x

1


,
…
,

x

n


)
↦

a

1



x

1


+
⋯
+

a

n



x

n


,


{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.

View More On Wikipedia.org
Recent contents Top users
  1. C

    I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

    I learned that for a bilinear form/square form the following theorem holds: matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form. Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
  2. PhysicsRock

    How to Show Linearity of a Function?

    I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
  3. whatevs

    Intro Math Are the (translated) High School Japanese maths textbooks by Kodaira Good?

    I'm trying to review some high school maths and work my way to Calculus and Linear Algebra, and I found these three translations of Japanese maths textbooks translated by the AMS and edited by Kunihiko Kodaira. The AMS links to them are: https://bookstore.ams.org/cdn-1669378252560/mawrld-8/...
  4. P

    Linear algebra problem with a probable typo

    Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and \begin{align} \left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
  5. PhysicsRock

    Prove relation between the group of integers and a subgroup

    So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt. $$ 0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
  6. PhysicsRock

    I Proof about pre-images of functions

    The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...
  7. A

    Prove the identity matrix is unique

    I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...
  8. J

    Linear operator in 2x2 complex vector space

    Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2? ____________________________________________________________ An ordered basis for C2x2 is: I don't...
  9. J

    Linear Transformation from R3 to R3

    "There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case? Thank you.
  10. murshid_islam

    What would be a good book for learning Linear Algebra by myself?

    Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)? I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
  11. J

    Is it possible to have a vector space with restricted scalars?

    I don't understand the solution: that for (1, ..., 1) the additive inverse is (-1, ..., -1), so the condition is not satisfied (and it is not a subspace). Which condition is not met? Thank you.
  12. MexChemE

    Analysis Study plan for Functional Analysis - Recommendations and critique

    Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
  13. guyvsdcsniper

    Linear Algebra in Dirac Notation

    I am trying to convert the attached picture into dirac notation. I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ> The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS. *Was going to type in LaTex but I...
  14. Graham87

    Intro to quantum mechanics - Spin and linear algebra

    So this expression is apparently in Sz basis? How can you see that? How would it look in Sy basis for example? The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here? Thanks
  15. Dario56

    I Inner and Outer Product of the Wavefunctions

    Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$ where $v_1,w_1...
  16. TGV320

    Courses Linear Algebra Self-Study: Textbook & Quantum Mechanics

    Hello, I have been looking for textbooks for self-studying linear algebra, which seems to be quite an important course. I have read that in order to study quantum mechanics well, one must have a very good command of linear algebra. Some textbooks in my country are quite bad and only teach...
  17. H

    Prove that every unitary matrix is diagonalisable by a unitary matrix

    Let's assume that ##A## is unitary and diagonalisable, so, we have ## \Lambda = C^{-1} A C ## Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##. I tried using some, petty, algebra to prove that ##C C* = I## but...
  18. T

    What are the two different senses of since and how are they used?

    Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus? Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
  19. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

    Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
  20. H

    Find a matrix ##C## such that ##C^{-1} A C## is a diagonal matrix

    I’m really unable to solve those questions which ask to find a nonsingular ##C## such that $$ C^{-1} A C$$ is a digonal matrix. Some people solve it by finding the eigenvalues and then using it to form a diagonal matrix and setting it equal to $$C^{-1} A C$$. Can you please tell me from scratch...
  21. H

    Proving ##(cof ~A)^t ~A = (det A)I##

    i-th column of ##cof~A## = $$ \begin{bmatrix} (-1)^{I+1} det~A_{1i} \\ (-1)^{I+2} det ~A_{2i}\\ \vdots \\ (-1)^{I+n} det ~A_{ni}\\ \end{bmatrix}$$ Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]## The I-th...
  22. H

    I How can I convince myself that I can find the inverse of this matrix?

    If I have a ##n\times n## matrix $$ U= \begin{bmatrix} u_{11} & u_{12} &u_{13} & \cdots u_{1n} \\ 0 & u_{22} & u_{23} & \cdots u_{2n} \\ 0&0 &u_{33} &\cdots u_{3n}\\ \vdots & \vdots &\vdots & \cdots \vdots \\ 0 & 0 & 0 &\cdots u_{nn} \end{bmatrix} $$ Now, I don't want to use the fact that it's...
  23. P

    Prove that ##S## is a subspace of ##V##

    Let ##S## be the subset of real (infinite) sequences (##a_1,a_2,\ldots##) with ##\lim a_n=0## and let ##V## be the space of all real sequences. Is ##S## a subspace of ##V##? Hello. I want to ask for help to start solving this problem. I don't understand how I can apply the theory I've studied...
  24. AimaneSN

    I Finding the orthogonal projection of a vector without an orthogonal basis

    Hi there, I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove : Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E## Then...
  25. H

    I How is uniqueness about the determinant proved by this theorem?

    Let me first list the four axioms that a determinant function follows: 1. ## d (A_1, \cdots, t_kA_k, \cdots, A_n)=t_kd(A_1, \cdots A_k, \cdots, A_n)## for any ##A_k## and ##t_k## 2. ##d(A_1, \cdots A_k + C , \cdots A_n)= d(A_1, \cdots A_k, \cdots A_n) + d(A_1, \cdots C, \cdots A_n)## for any...
  26. H

    I The way matrices are written without boxes

    When a matrix is represented as a box it seems all very clear, but this representation $$ A = (a_{ij} )_{i, j =1}^{m,n}$$ Isn't very representative to me. The i -j thing creates a lot of confusion, when we write ##a_{ij}## do we mean the element of i th row and jth column or the other way...
  27. A

    Engineering Signals & Systems with Linear Algebra

    Hello everyone, I would like to get some help with the above problem on signals and linear projections. Is my approach reasonable? If it is incorrect, please help. Thanks! My approach is that s3(t) ad s4(t) are both linear combinations of s1(t) and s2(t), so we need an orthonormal basis for the...
  28. H

    Prove that T is a linear transformation

    We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, : Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?) OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...
  29. H

    Prove that ##\langle x, y \rangle = 0 \iff ||x + cy|| \geq ||x||##.

    (We are working in a real Euclidean space) So, we have to show two things: (1)the arrow goes from left to right, (2) the arrow comes from right to left. (1) if we're given ##\langle x, y \rangle = 0 ## $$ || x+ cy||^2 = \langle x,x \rangle + 2c\langle x,y\rangle +c^2 \langle y,y \rangle $$ $$...
  30. H

    How to convince myself that I can take n=1 here?

    The Homework Statement reads the question. We have $$ \langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right) $$ If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows $$...
  31. Vectronix

    Linear Algebra I need a book on linear algebra....

    Is Advanced Linear and Matrix Algebra by Nathaniel Johnston a good book on linear algebra? Will it teach me all I need to know? Is there any calculus in it despite the name? I never took a course on linear algebra so I'm looking for something that teaches everything and includes calculus with...
  32. terrytosh

    Should I study Analysis before Linear Algebra?

    Or is reading a proofs book enough
  33. H

    A quite verbal proof that if V is finite dimensional then S is also....

    If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##. Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as $$ x =...
  34. S

    I Linear Algebra 1 problem, Vector Geometry: Lines

    Problem: Given the line L: x = (-3, 1) + t(1,-2) find all x on L that lie 2 units from (-3, 1). I know the answer is (3 ± 2 / √5, -1 ± 4/√5) but I don't know where to start. I found that if t=2, x= (-5, 5) and the normal vector is (2, 1) but I am not sure if this information is useful or how...
  35. H

    I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?

    ##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...
  36. L

    I How can I find all possible Jordan forms?

    Hi this is my first message in this forum , I have this problem in my linear algebra course and I have never seen this type. Let $T : \mathbb{Q}^3 → \mathbb{Q}^3 $ a linear application s.t $(T^7 + 2I)(T^2 + 3T + 2I)^2 = 0$ Find all possible Jordan forms and the relative characteristic...
  37. M

    Algebra Looking for my first textbook on Linear Algebra Need suggestions

    First of all, I attached pictures of the very last algebra textbook that I have finished studying. I'm going the self taught route. I really loved this book because it had lots of examples, practice exercises, quizzes and even tests! It also had answers in the back. It's currently my favorite...
  38. L

    I Prove that the limit of this matrix expression is 0

    Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that: $$ \lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0 $$ where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...
  39. JD_PM

    Statements about linear maps | Linear Algebra

    First thing to notice is that ##L## and ##L \circ L## are precisely equal linear maps. What we know $$L \ \text{is injective} \iff \ker(L)=\{0\}$$ $$\ker L' = \{ x \in \Im(L) \ | \ L'(x)=0\}$$ $$\Im(L)=\{ x \in V \ | \ \exists \ v \in V \ \text{such that} \ L(v)=x\}$$ Besides, we notice...
  40. S

    I General Method for Mapping an Ellipsoid to Unit Sphere

    I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any...
  41. Alwar

    I How to Find the Generalized Eigenvector in a Matrix ODE?

    Hi, I have a set of ODE's represented in matrix format as shown in the attached file. The matrix A has algebraic multiplicity equal to 3 and geometric multiplicity 2. I am trying to find the generalized eigenvector by algorithm (A-λI)w=v, where w is the generalized eigenvector and v is the...
  42. S

    I Proving (x,x) = 0 implies x = 0 in real vector space V

    I have the followinq question: Let ##(,)## be a real-valued inner product on a real vector space ##V##. That is, ##(,)## is a symmetric bilinear map ##(,):V \times V \rightarrow \mathbb{R}## that is non-degenerate Suppose, for all ##v \in V## we have ##(v,v) \geq 0## Now I want to prove that...
  43. JD_PM

    Finding a complementary subspace ##U## | Linear Algebra

    We only worry about finite vector spaces here. I have been taught that a subspace ##W## of a vector space ##V## has a complementary subspace ##U## if ##V = U \oplus W##. Besides, I understand that, given a finite vectorspace ##(\Bbb R, V, +)##, any subspace ##U## of ##V## has a complementary...
  44. JD_PM

    Given subspaces ##U \& W##, show they are equal | Linear Algebra

    Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal. I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would...
  45. A

    Linear Algebra - LU Factorization

    Hello all, I have a problem related to LU Factorization with my work following it. Would anyone be willing to provide feedback on if my work is a correct approach/answer and help if it needs more work? Thanks in advance. Problem: Work:
  46. JD_PM

    Proving statements about matrices | Linear Algebra

    Hi guys! :) I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected) I think this one is true, as there is no squared...
  47. Lecture 5 - Science, Toys, and the PCA

    Lecture 5 - Science, Toys, and the PCA

    We open this lecture with a discussion of how advancements in science and technology come from a consumer demand for better toys. We also give an introduction to Principle Component Analysis (PCA). We talk about how to arrange data, shift it, and the find the principle components of our dataset.
    • AcademicOverAnalysis
    • Media item
    • Data science Linear algebra Pca
    • Comments: 0
    • Category: Misc Math
  48. Lecture 3 - How SVDs are used in Facial Recognition Software

    Lecture 3 - How SVDs are used in Facial Recognition Software

    This video builds on the SVD concepts of the previous videos, where I talk about the algorithm from the paper Eigenfaces for Recognition. These tools are used everywhere from law enforcement (such as tracking down the rioters at the Capitol) to unlocking your cell phone.
    • AcademicOverAnalysis
    • Media item
    • Data science Linear algebra Svd
    • Comments: 0
    • Category: Misc Math
  49. Lecture 2 - Understanding Everything from Data - The SVD

    Lecture 2 - Understanding Everything from Data - The SVD

    In this video I give an introduction to the singular value decomposition, one of the key tools to learning from data. The SVD allows us to assemble data into a matrix, and then to find the key or "principle" components of the data, which will allow us to represent the entire data set with only a few
    • AcademicOverAnalysis
    • Media item
    • Data science Linear algebra Pca Svd
    • Comments: 0
    • Category: Misc Math
  50. A

    Riesz Basis Problem: Definition & Problem Statement

    The reference definition and problem statement are shown below with my work shown following right after. I would like to know if I am approaching this correctly, and if not, could guidance be provided? Not very sure. I'm not proficient at formatting equations, so I'm providing snippets, my...
Back
Top