Chapter 8 - Independence, Basis, and Dimension for Subspaces of (R^n)

Section 8.2 -  Independent and Dependent Sets of Vectors in (R^n)

Example 8.2.1

A={{5,-7,1,4,-9,3},{-2,-1,6,0,-5,8},{11,9,-3,2,5,7}};

(R=RowReduce[A])//MatrixForm

Row[{Transpose[{Join[-R[[All,4]],{1,0,0}]}]//MatrixForm,Transpose[{Join[-R[[All,5]],{0,1,0}]}]//MatrixForm,Transpose[{Join[-R[[All,6]],{0,0,1}]}]//MatrixForm}]

Example 8.2.2

u={5,-7,3,1,-2};v={-9,0,8,-4,6};

w=-3 u+10 v

RowReduce[{u,v,w}]//MatrixForm

RowReduce[{v,w,u}]//MatrixForm

RowReduce[{w,v,u}]//MatrixForm

Example 8.2.3

z=2 u-7 v

(T=RowReduce[{w,u,z,v}])//MatrixForm

(L=RowReduce[{u,v}])//MatrixForm

Example 8.2.5

A={{1,-3,2,5,-6,4,0},{-9,2,-1,7,1,8,0}};

RowReduce[A]//MatrixForm

Clear[x,y,z,u,v,w]

X={x,y,z,u,v,w};

U=Drop[A[[1]],-1];V=Drop[A[[2]],-1];

Solve[{X.U==0,X.V==0},{x,y}]

(Subscript[w, 1])={1,17,25,0,0,0};

(Subscript[w, 2])={31,52,0,25,0,0};

(Subscript[w, 3])={-9,-53,0,0,25,0};

(Subscript[w, 4])={32,44,0,0,0,25};

{U.(Subscript[w, 1]),U.(Subscript[w, 2]),U.(Subscript[w, 3]),U.(Subscript[w, 4])}

{V.(Subscript[w, 1]),V.(Subscript[w, 2]),V.(Subscript[w, 3]),V.(Subscript[w, 4])}