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.

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[[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...

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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. 

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