Vector Spaces

Vector Spaces Assignment

Problem 1 Problem 2

a )b

Steinitz Replacement Theorem B ' is a basis of V3 (R B ' is a basis of V3 (R Problem 3

Particular Example X and Y are linearly independent

X and Y are bases for V2 (R ) which is a subspace of V3 (R

a x1c x2d where x1 2 , x2

b y1c y2d where y1 -1 , y2

b (y1c y2d

y2 )d )d

General Form

Let X a , b and Y c , d both be bases for subspace

S of vector space V

a x1c x2d since a is in subspace S , and c ,d is a basis for S

b y1c y2d since b is in subspace S , and c ,d is a basis for S

is the coordinate vector of a with respect to basis Y

is the coordinate vector of b with respect to basis Y

Let u be any vector is subspace S

b since u is in subspace S , and a ,b is a basis for S

is the coordinate vector of u with respect to basis X (y1c y2d

y2 )d

is the coordinate vector of u with respect to basis Y is the transition matrix of any u in S from basis X to Y

The transition matrix

is formed by cascading the coordinate vectors of the basis vectors of basis X with respect to basis Y PAGE

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Vector Spaces Assignment...