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 w + vector x, vector x - vector w, vector u - vector v + vector w
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By the given problem the vectors are 2u-v, w+x, x-w, u-v+w. To check the linear dependence or independence we prepare the matrix corresponding to these vectors.
which can be rearranged to the matrix
We see that determinant of the last matrix is non zero. So the given four vectors (2u-v),(w+x),(x-w), (u-v+w) are linearly independent.
Let vect a=2u-v
Let vectors a,b, c adn d are linearly independent . By def of L.I. if
pa+qb+rc+sd=0 then p=q=r=s=0
`` Write row echlon form of the coefficient matrix ,we have
Thus rank of the coefficient matrix is 4.
so only solution we have
Thus vectors a,b,c and d are linearly independent.
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