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linearAlgebra1

# linearAlgebra1 - Linear Algebra Basics A vector space(or...

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Linear Algebra Basics A vector space (or, linear space) is an algebraic structure that often provides a home for solutions of mathematical models. Linear algebra is a study of vector spaces and a class of special functions (called “linear transformations”) between them. The terms vector space and linear space are interchangeable. We’ll use them both. 1. Linear Spaces : The elements of a linear space are called vectors. We can scale and add vectors. By scaling and adding vectors we can build new vectors. This simple construction paradigm is remarkably useful. Definition : A linear space (or vector space ) is a set L together with field of scalars F and two functions (operations); addition + : L × L L and (scalar) multiplication · : F × L L satisfying conditions (a)-(h) below. For us, the scalar field F will almost always be the real numbers R (rarely, for us, it may be the complex numbers C ). Also, for brevity (clarity?), if α F and x L and we want to scale x using α , then we’ll write αx rather than α · x . (a) x + y = y + x for x , y in L (b) x + ( y + z ) = ( x + y ) + z for x, y, z L (c) there is an element O L such that for each x L , x + O = x (d) for each x L there corresponds y L such that x + y = O (e) for each α, β F and x L , α ( βx ) = ( αβ ) x (f) for each α, β F and x L , ( α + β ) x = αx + βx (g) for each α F and x, y L , α ( x + y ) = αx + αy , and (h) 1 x = x for each x L . Remarks : If x + y = x + z then y = z . This cancellation law holds because y = O + y = ( - x + x ) + y = - x + ( x + y ) = - x + ( x + z ) = ( - x + x ) + z = O + z = z . 0 x + x = (0 + 1) x = x so 0 x = O . The zero element O of (b) and y , the additive inverse of x , in (c) are unique; we usually write - x for the additive inverse of x . Since - 1 x + x = - 1 x + 1 x = ( - 1 + 1) x = 0 x = O then - x = - 1 x . For us, the scalar field F will always be either the real numbers R or complex numbers C . There are other fields however, for example the rational numbers Q . There are finite fields, e.g., Z p (the integers mod p ) where p is a prime number. Vector spaces over finite fields find applications in, for example, crytography. We’ll not consider them in this course. Examples of linear spaces: (a) R n : A vector x R n is an n -tuple of real numbers, x = ( x 1 , x 2 , . . . , x n ). The set of real numbers R is the scalar field. Vector additon and scalar multiplication are defined component-wise: if x and y are vectors and α R then x + y = ( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) := ( x 1 + y 1 , x 2 + y 2 , . . . , x n + y n ) and αx = α ( x 1 , x 2 , . . . , x n ) := ( αx 1 , αx 2 , . . . , αx n ). (b) C n : A vector z C n is an n -tuple of complex numbers, z = ( z 1 , z 2 , . . . , z n ). The set of complex numbers C is the scalar field. Vector addition and scalar multiplication are defined component-wise: if w and z are vectors and α C then w + z = ( w 1 , w 2 , . . . , w n ) + ( z 1 , z 2 , . . . , z n ) := ( w 1 + z 1 , w 2 + z 2 , . . . , w n + z n ) and αz = α ( z 1 , z 2 , . . . , z n ) := ( αz 1 , αz 2 , . . . , αz n ). (c) R n p : If p R n then define R n p := { ( p, x ) | x R n } . R n p (vectors based at p ) becomes a real vector space if addition and scalar multiplication are defined by ( p, x ) + ( p, y ) = ( p, x + y ) and α ( p, x ) = ( p, αx ) for each scalar α R and pair of vectors x, y R n .

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(d) M m × n : A vector is a rectangular m × n array of real numbers.
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linearAlgebra1 - Linear Algebra Basics A vector space(or...

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