Unit 1

 (net time: 78 min)

The contents learned in MATH270 is not included.

4.7 Row/Column/Null Space

Theorems:

Ax=b is consistent ó b is in the column space of A

The solution of Ax=b can be expressed as the sum a particular solution of Ax=b and the general solution of Ax=0

Elementary row operations

1. do not change the null space (solution space of Ax=0) or the row space.
2. do not change the dimension of column space.

so, dim(row(A)) =dim(col(A)) =rank(A)

Methods:

Row echelon form to get the basis of

1. row space: use result directly
2. column space:

use the position of result

the dependency equation can also be copied.

denoted by vectors of smaller subscripts

1. row space with original form: turn to column space, change to row echelon form and use the position

4.8 Rank/Nullity

Theorems:

nullity: dim(null(A)) = number of parameters in Ax=0

rank(A) = number of leading variables <= min (m, n), given A an m * n matrix

rank(A) + nullity(A) =n, given A with n column

10.9 Computer Graphics

coordinate matrix: 3 * n

scaling: multiply a diagonal matrix

translation: add a matrix will all columns the same

rotation: trigonometry

stereoscopic pair: rotate + translate