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...

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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rakesh05's profile pic

rakesh05 | High School Teacher | (Level 1) Assistant Educator

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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.

``

[[2,-1,0,0],[0,0,1,1],[0,0,-1,1],[1,-1,1,0]]

which can be rearranged to the matrix

[[1,-1,1,0],[2,-1,0,0],[0,0,1,1],[0,0,-1,1]]~[[1,-1,1,0],[0,-1,2,0],[0,0,1,1],[0,0,0,2]]

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. 

``

pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

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Let vect a=2u-v

            b=w+x

            c=-w+x

and      d=u-v+w

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

`[[2,-1,0,0],[0,0,1,1],[0,0,-1,1],[1,-1,1,0]][[p],[q],[r],[s]]=[[0],[0],[0],[0]]`

`` Write row echlon form of the coefficient matrix ,we have

`[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,0,1]][[p],[q],[r],[s]]=[[0],[0],[0],[0]]`

Thus rank of the coefficient matrix is 4.

so only solution we have

p=q=r=s=0

Thus vectors a,b,c and d are linearly independent.

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