# Linear Algebra (LA-101A)

Contents:

CH-1-Vectors

$\bullet \text{ Perpendicularity } \; \mid \bullet \text{ Scalar Product } \; \mid \bullet \text { Norm of a vector } \; \mid \bullet \textbf { Triangle Inequality }$

CH-2-Vector Spaces

$\; \bullet \text { Field } \; \mid \bullet \text { Vector space } \; \mid \bullet \text{ Subspace } \; \mid \bullet \text { function space } \; \mid \bullet \text { Basis } \; \mid \bullet \text { Linear Independence } \; \mid \bullet \text{ Co-ordinates } \; \mid \bullet \text { Dimension of vector space } \; \mid \bullet \text { maximal set of linear independent elements of vector space } \; \mid \bullet \text { Direct sums } \; \mid \bullet \text{ Direct products }$

CH-3-Matrices

$\; \bullet \text { Linear Equations(L.E.)} \; \mid \bullet \text { Homogeneous system of L.E. } \; \mid \bullet \text{ Invertible/Singular Matrix } \; \mid \bullet \text { Uniqueness of Inverse Matrix } \; \mid \bullet \text { Unit Matrix } \; \bullet \text { Transpose of a matrix }$

CH-4-Linear Mappings

$\; \bullet \text { Mappings } \; \mid \bullet \text { Co-ordinate functions } \; \mid \bullet\text{ Parametric curve in n-space } \; \mid \bullet\text { Composite mappings } \; \mid \bullet \text { Injective mapping } \; \mid \bullet \text { Surjective Mappings } \; \mid \bullet \text { inverse mappings } \; \mid \bullet \text { translation mappings } \; \mid \bullet \text { linear mappings } \; \mid \bullet \text{ K-linear map } \; \mid \bullet \text { Identity and Zero mappings } \; \mid \bullet \text { set of linear maps } \; \mid \bullet \text { Parallelogram spanned by bases } \; \mid \bullet \textbf{ dim(V)=dim[Im(f)]+dim[Ker(f)] } \; \mid \bullet \text { Kernel of Linear map } \; \mid \bullet \text { Kernel is subspace } \; \mid \bullet \text { composition and inverse of linear map } \; \mid \bullet \textbf { Isomorphism } \; \mid \bullet \text { Geometric application } \; \mid \bullet \text { Parallelogram } \; \mid \bullet \text { Triangle } \; \bullet \textbf{ Convex set }$

CH-5-Linear Maps and Matrices

$\; \bullet \text { Linear Maps associated with a matrix } \; \mid \bullet \text { Matrix associated with linear maps } \; \mid \bullet \text{ Projection Linear map } \; \bullet \text { Rotation Operation } \; \mid \bullet \text { Rotation preserves norms } \; \mid \bullet \text { Bases,Matrices, Linear Maps } \; \mid \bullet \text{ Isomorphism between space of linear maps and matrices } \; \mid \bullet \text { Change of bases } \; \mid \bullet \text { Diagonalizable matrix } \; \mid \bullet \text { Similar matrix }$

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Books: Linear Algebra by Serge Lang (2nd edition)