Span

[hkus10]

Determine whether the given vector A in 2x2 matrix belongs to span{A1, A2, A3}, where

A1 =

[1 -1

0 3]

 

A2 =

[1 1

0 2]

 

A3 =

[2 2

-1 1]

 

A =

[5 1

-1 9].

 

Since A1, A2, A3 are not a nx1 matrices, I cannot put this into reduced echelon form? Therefore, what can I do to solve this problem?

 

Thanks

[timo]

You write down a condition for A being an element of span{A1,A2,A3} and see if the resulting equations can be satisfied.

[acidhoony]

you may got c1A1 + c2A2 + c3A3 = (a b)

(c d)

 

which a,b,c,d are arbitary real number maybe~

 

if you can find c1,2,3for every a,b,c,d

 

then it A1,2,3 is span all 2 by 2 Matrix.

 

but i think it will not span.

 

Because a,b,c,d is fourrr v variable but so we need at least for basis

 

but we have only A1,2,3 ; only three ;;; i don;t know it is dependent or not , but , whatever it is

 

it is too little number of matrix to span every two by two matrix~

[the tree]

The simplest, crudest approach is to write it as a set of linear equations and look for a solution from there.

 

If [imath]c_1 A_1 + c_2 A_2 + c_2 A_2 = A[/imath], then that implies four equations that can be reduced using good old fashioned elimination.

 

The more mathsy approach would be to summarise the the set described by [imath]\mbox{span}( A_1 , A_2 , A_3 )[/imath] in such a way that it's obvious whether or not [imath]A[/imath] is included. As Acidhoony said, that set is going to be at most 3-dimensional.

Edited March 16, 2011 by the tree