Tuesday, 19 November 2013

linear algebra - Find a basis of the subspaces of mathbbR4 generated by the vectors



Find a basis of the subspaces of R4 generated by the vectors v1=(1,1,2,0),v2=(1,0,1,0),v3=(2,2,0,0),v4=(0,0,1,2)



First of all I wrote these vectors as rows of a matrix then applied the following transformations to reduce the matrix in row echelon form:-
R2+R1 & R32R1

Then R3+4R2 and finally 110R3
Then I wrote the non zero rows in row echelon form as B=(1,1,2,0),(0,1,3,0),(0,0,1,0) which forms basis .
Am I right here?


Answer



Using the vectors given in the problem, we can define a set: A={v1,v2,v3,v4}



Remember the definition of a Basis set:



Given: V is a vector space and B is a subset of V, we say that B is a basis of V IFF V=span(B) and B is linearly independent.




Row-Reducing the matrix form of the set A yields:
[1120102021010002]


<=>

[1000010000100001]

Since each column contains a leading "1", we can say that the set A is linearly independent.



This result implies that the set A itself satisfies the definition of a basis.



B={v1,v2,v3,v4}


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...