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Math - 1 I VECTOR SPACE Consider a two dimensional plane...

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1 I. VECTOR SPACE Consider a two dimensional plane, every point in the place can be mapped to a unique vector. All of the vectors form a linear vector space v with a dimensionality equal to two. Any vector can be labeled as ( x, y ) which can be written as ( x, y ) = x (1 , 0) + y (0 , 1) (1) The sum of two vectors satisfies, a ( x 1 , y 1 ) + b ( x 2 , y 2 ) = ( ax 1 + bx 2 + ay 1 + by 2 ) (2) In this vector space, we can only define a dot product, ( x 1 , y x ) · ( x 2 , y 2 ) = ( x 1 x 2 , y 2 y 2 ) (3) Notice that (1 , 0) · (0 , 1) = 0 and (1 , 0) · (1 , 0)=1, which we call the two vectors are orthogonal to each other. Since any vector can be written as a linear combination of the two orthogonal vectors, we call the linear vector space v has dimension two and (1 , 0) and (0 , 1) form a complete orthonormal basis of V . II. DIRAC NOTATION The similar idea can be generalized to a space with any dimensions. We call a general linear vector space V with dimension N if any vector in V can be written as a linear combination of N orthonormal vectors. We will use a ket

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Math - 1 I VECTOR SPACE Consider a two dimensional plane...

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