B.Sc Method Chapter 6 (Vector Spaces) Notes

Mathematical Methods (B.Sc.)

Chapter:06 (Vector Spaces)

In this chapter, we will study the following topics:

• definition and examples
• scalars and vectors
• the zero vector
• real or complex vector space
• distributive law in F
• distributive law in F
• associative law in F
• the identity element of F
• n-dimensional Euclidean space
• subspaces
• the sum of two subspaces
• smallest subspace
• direct sum
• a linear combination and spanning set
• finite dimensional
• infinite dimensional
• linearly independent
• linear combination
• the standard basis for R
• the space of 3×3 symmetric matrices
• the space of 4×4 symmetric matrices
• the space of nxn symmetric matrices
• row and column spaces
• inverse elementary row operation
• column rank
• the inverse elementary row operation
• rank-alternative method
• consistency criterion
• solution space
• linear transformation
• nonsingular transformation
• A one-to-one linear transformation preserves basis and dimensions
• a matrix of a linear transformation
• coordinate vector

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