Editing 01.25.06

mportant Linear Algebra Concepts:

'''1. Operations on vectors.'''

An n-dimentional cartesian space is <m>bbR^n = delim {lbrace} {(x_1,cdots,x_n) vert x_i in bbR} {rbrace}</m>

Addition, <m>+:bbR^n*bbR^n right bbR^n</m>, is performed componentwise. If <m>a=(a_1,cdots,a_n)</m>
and <m>b=(b_1,cdots,b_n)</m> then <m>a+b=(a_1+b_1,cdots,a_n+b_n)</m>. Geometrically, a parallelogram rule is used.

Scalar multiplication, <m>*:bbR*bbR^n right bbR^n</m>
is again done componentwise. If <m>c in bbR</m> is a scalar and <m>v=(v_1,cdots,v_n)</m> is a vector then <m>c*v=(c*v_1,cdots,c*v_n)</m>. Geometrically, the length is scaled by a factor of <m>delim{|}c{|}</m>.
The direction of<m>c*v</m>is the same if <m>c>0</m> and opposite if <m>c<0</m>.

<m>(bbR^n,+,*)</m> is a real vector space.

'''2. Linear Combinations.'''

If <m>v_1,cdots,v_m</m> are vectors, and <m>c_1,cdots,c_m</m> are scalars, then <m>c_1*v_1+cdots+c_n*v_n</m> is called a linear combination. If <m>X in bbR^n</m> is closed under linear combinations then <m>X</m> is called a linear subspace of <m>bbR^n</m>.

'''3. Basis of a linear subspace.'''

If <m>v_1,cdots,v_n</m> are linearly independent vectors and <m>V</m> is a linear subspace consisting solely of <m>v_1,cdots,v_n</m> and all their linear combinations then we call <m>B = lbrace v_1,cdots,v_n rbrace</m> a basis of <m>V</m>. Any vector <m>v in V</m> is then uniquely expressible as a linear combination of <m>v_1,cdots,v_n</m>. If <m>v=c_1*v_1,cdots,c_n*v_n</m> denote its representation in <m>B</m> by <m>delim{[}{v}{]}_B=(c_1,cdots,c_n)</m>. All bases of <m>V</m> have the same number of vectors. We call that number <m>dim(V)</m>. The vectors <m>e_1=(1,0,0,cdots,0), e_2=(0,1,0,cdots,0), cdots, e_n=(0,0,0,cdots,1)</m> make up a standard basis of <m>bbR^n</m>.

'''4. Linear transformations'''
<m>T: bbR^n right bbR^m</m> is called a linear transformation if for any two vectors <m>v_1,v_2 in bbR^n</m> and scalars <m>c_1, c_2 in bbR</m>, <m>T(c_1*v_1+c_2*v_2) = c_1*T(v_1)+c_2*T(v_2)</m>. Given a basis <m>B</m> of <m>bbR^n</m> and <m>B prime</m> of <m>bbR^m</m> then a linear transform T can be expressed go from <m>B</m> to <m>B prime</m> as <m>delim{[}{T}{]}_{B,B prime} = delim{[}{delim{[}{T(v_1)}{]}_{B prime},cdots,delim{[}{T(v_n)}{]}_{B prime}}{]}</m>. In general <m>delim{[}{Id}{]}_{B,B} = I_n</m> but <m>delim{[}{Id}{]}_{B,B prime} <> I_n</m>

'''5. Multilinear transforms.'''

Let <m>V</m> be a vector space. A function <m>f:V^n right F</m> is called multilinear if for <m>i=1,2,cdots,n</m> we have <m>cf(v_1,cdots,v_i,cdots,v_n) = f(v_1,cdots,cv_i,cdots,v_n)</m> and
<m>f(v_1,cdots,v_i,cdots,v_n) + f(v_1,cdots,v_i prime,cdots,v_n) = f(v_1,cdots,v_i+v_i prime,cdots,v_n)</m>.

Examples are determinants, dot-products, and cross-products.