Assume that vector u, vector v, vector w, vector x are linearly independent. Determine whether the following vectors are linearly independent or dependent. If they are linearly dependent then write a dependence relation for them.
2(vector u)-vector v + vector x, 3(vector v) + vector w + 4(vector x), vector x + vector w, 2(vector u) -2(vector v) + vector w + vector x
If we prepare the matrix corresponding to the above set of vectors we get
which can be rearranged to
Finally we see that determinant of the above matrix is non zero. And so vectors are linearly independent.
Let vect a=2u-v+0w+x
Let p,q,r,s be real nos, such that
then p=q=r=s=0 .
Then vectors a ,b, c, and d are Linearly independent otherwise linearly dependent.
`` write row echlon form of coefficient matrix
`[[1,0,0,1],[0,1,0,1],[0,0,1,1],[0,0,0,0]][[p],[q],[r],[s]]=[,,,]` ,thus rank of the coefficient matrix is 3.
There are infinite numbers of value of p,q,r and s. So vectors a,b,c, and d are linearly dependent.
thus we assume `s!=0` , p+s=0 , q+s=0, q+r=0
solving syestem of equations,we have
p=-s,q=-s and r=s .
Thus from (i) ,we have
d=a+b-c which is the dependence relation