01.25.06

From Vectorcalcumb

mportant Linear Algebra Concepts:

1. Operations on vectors.

An n-dimentional cartesian space is <m>bbR^n = 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. 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.