j3259の日記： 線形代数

v = [v1
     v2]

v = [v1  and w = [w1   and v + w = [v1 + w1
     v2]          w2]               v2 + w2]

kv = [kv1
      kv2]

The dot product or inner product of v = (v1, v2) and w = (w1, w2) is the number

v DOT w = v1w1 + v2w2

The length (or norm) of a vector is the square root of v DOT v:

length = |v| = sqrt(v DOT v)

A unit vector u is a vector whose length equals one. Then

u DOT u = 1

1A: Divide any nonzero vector v by its length. Then

u = v / |v|

is a unit vector in the same direction as v.

1B: The dot product v DOT v is zero when v is perpendicular to w.

1C: a) If u and U are unit vectors then

u DOT U = cos(theta)

b) If u and U are unit vectors then |u DOT U|

2C: The identity matrix has 1's on the diagonal and 0's everywhere else. Then Ib = b. The elementary matrix or elimination matrix Eij that substracts a multiple l of row j from row i has the extra nonzero entry -l in the i, j position.

I = [1 0 0  and E31 = [1 0 0
     0 1 0             0 1 0
     0 0 1]           -l 0 1]

Pivot is the first nonzero in the row. To solve n equations we want n pivots.

A = [a11 a12  and B = [b11 b12   and AB = [(a11b11 + a12b21) (a11b12 + a12b22)
     a21 a22]          b21 b22]            (a21b11 + a22b21) (a21b12 + a22b22)]

Let the blocks of A be its columns. Let the block of B be its rows. Then block multiplication AB is columns times rows.

The matrix A is invertible if there exists a matrix A^-1 such that

A^-1A = I and AA^-1 = I.

The Gauss-Jordan method computes A^-1.

[A e1 e2 e3] = [2 -1  0 1 0 0
               -1  2 -1 0 1 0
                0 -1  2 0 0 1]

            => [2 -1   0 1   0 0
                0 3/2 -1 1/2 1 0   (1/2 row 1 + row 2)
                0 -1   2 0   0 1]

            => [1 0 0 3/4 1/2 1/4
                0 1 0 1/2  1  1/2
                0 0 1 1/3 2/3  1]

A^T, the transpose of A uses the rows of A as its columns.

If A = [1 2 3  then trans(A) = [1 0
        0 0 4]                  2 0
                                3 4]

An n by n permutation matrix P has the n rows of I in any order. There are n! permutation matrices of order n.

The space Rⁿ consists of all column vectors v with n components.

A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: If v and w are vectors in the subspace and c is any scalar, then

1. v + w is in the subspace and
2. cv is in the subspace

The column space of A consists of all linear combination of the columns. The combinations are all possible vectors Ax.

The system Ax = b is solvable if and only if b is in the column space of A.

The nullspace of A consists of all solutions to Ax = 0. These vectors x are in Rⁿ. The nullspace containing all solutions to x is denoted by N(A).

http://en.wikipedia.org/wiki/Determinants

det A = sum over all n! permutations P = (alpha, beta, ..., omega)
= sigma(det p)a_1alphaa_2beta...a_{n omega}

|a b  = ad - bc
c d|

|a11 a12 a13  = +a11a22a33 + a12a23a31 + a13a21a32
a21 a22 a23    -a11a23a32 - a12a21a33 - a13a22a31
a31 a32 a33|

plus "down right" minus "down left".

5D: The triangle with corners (x1, y1) and (x2, y2) and (x3, y3) has area = 1/2(determinant):

area = 1/2 |x1 y1 1  or area = 1/2 |x1 y1  when (x3, y3) = (0, 0)
            x2 y2 1                 x2 y2|
            x3 y3 1|

The cross product of u = (u1, u2, u3) and v = (v1, v2, v3) is the vector

u CROSS v = |i  j  k   = (u2v3 - u3v2)i + (u3v1 - u1v3)j + (u1v2 - u2v1)k
             u1 u2 u3
             v1 v2 v3|

This vector is perpendicular to u and v. The cross product v CROSS u is -(u CROSS v).

The cross product is a vector with |u| |v| |sin(theta)|. Its direction is perpendicular to u and v. It points "up" or "down" by the right hand rule.