Shahada ya kwanza (mwanzo)AlgebraImepangwa

Linear algebra

Systems of linear equations and Gaussian elimination, matrices, vector spaces, linear maps, bases and dimension, rank–nullity, determinants, eigenvalues and diagonalisation.

math.algebra.linear-algebra

Onyesha kila kinachohitajika kufikia mada hii

Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.

Inajengwa juu ya

Linear relationships and simultaneous equations Real numbers and irrationality

Masharti ya awali ya lazima 20 (hadi chini kabisa)

Utajifunza

Solve linear systems by Gaussian elimination and describe solution sets as translates of null spaces.
Prove that all bases of a finite-dimensional vector space have the same size, and apply the rank–nullity theorem.
Represent linear maps by matrices, compose them by matrix multiplication and change basis.
Compute eigenvalues and eigenvectors and decide whether a matrix is diagonalisable.

Fasili

A vector space over a field k is a set with addition and scalar multiplication satisfying the vector-space axioms.
A linear map T : V → W satisfies T(u + v) = T(u) + T(v) and T(cv) = cT(v).
A basis is a linearly independent spanning set; dim V is the number of elements in a basis.
λ is an eigenvalue of T if Tv = λv for some non-zero vector v.

Nadharia

Rank–nullity: dim V = dim ker T + dim im T for a linear map T : V → W with V finite-dimensional.
A composite of linear maps is linear, and corresponds to the product of their matrices.
A square matrix is invertible if and only if its determinant is non-zero, if and only if its columns are linearly independent.
Eigenvectors belonging to distinct eigenvalues are linearly independent.

Mawazo potofu ya kawaida

Thinking matrix multiplication is commutative.
Assuming every square matrix is diagonalisable; the matrix with rows (0, 1) and (0, 0) is not.
Believing a linear map must send lines through the origin to lines; it may collapse them to a point.

Inaelekea kwa

Transformation geometry and the Erlangen programme Multivariable and vector calculus Ordinary differential equations Fields and Galois theory Advanced linear algebra Markov chains Numerical analysis Optimisation Mathematical biology

Ugunduzi wa hiari

Categories Vector spaces and linear maps form a category: composites of linear maps are linear and identity maps are linear.
Representation theory of finite groups A representation of a group is a homomorphism into the group of invertible linear maps of a vector space.
← Linear equations and inequalities
← The coordinate plane

Njia za kujifunza

Usomaji zaidi

Sheldon Axler, Linear Algebra Done Right (1996)
Gilbert Strang, Introduction to Linear Algebra