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.

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  1. Hill

    A gardener collected 17 apples...

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  2. TGV320

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  3. Z

    Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1

    This is problem 28 from chapter 3F "Duality" of Axler's Linear Algebra Done Right, third edition. I spent quite a long time on this problem, like a few hours. Since there is no available solution, I am wondering if my solution is correct. One assumption in this problem is that...
  4. Z

    Prove that range ##T'## = ##(\text{null}\ T)^0##

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  5. Z

    Given surjective ##T:V\to W##, find isomorphism ##T|_U## of U onto W.

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  6. Z

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  7. Infrared

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  8. GGGGc

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  9. D

    I Dimension and solution to matrix-vector product

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  10. Z

    Decompose 4x4 determinant into 24 determinants -- How many are zero?

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  11. crememars

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  12. D

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  13. T

    B Question on basic linear algebra (new to the subject)

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  14. giodude

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  15. chwala

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  16. chwala

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  17. S

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  18. S

    I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'

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  19. rajsekharnath

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  20. Infrared

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  21. C

    Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

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  22. M

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  23. M

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  24. C

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  25. C

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  26. C

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  27. A

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  28. A

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  29. F

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  31. C

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  32. hilbert2

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  33. H

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  34. Vanilla Gorilla

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  36. S

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  38. S

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  39. C

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  40. dubeypuja

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  41. D

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  43. PhysicsRock

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  44. whatevs

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  45. P

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  47. PhysicsRock

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  48. A

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  49. J

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  50. J

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