Prove that for any positive integers m and n, thecheck digit code in with check vector c 1 [1, 1, . . . , 1] will detect all single errors. (That is, provethat if vectors u and v in differ in exactly one entry,then c # u c # v. 2

1.7 Linear Independence n - An indexed set of vectors {v ,…,1 } ip R is said to be linearly independent if the vector equation + + ⋯+ = 0 has only the trivial solution. ▯ ▯ ▯ ▯ ▯ ▯ The set {v 1…,v p is said be linearly dependent if their exist weights c 1…,c p not all zero, such that ▯ ▯+ ▯ ▯ ⋯+ =▯ ▯ - + + ⋯+ = 0 à linear dependence relation amount v ,…,v , when ▯ ▯ ▯ ▯ ▯ ▯ 1 p the weights are not all zero Linear Independence of Matrix Columns - The columns of a matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution Sets of One or Two Vectors - A set containing only one vector is linearly independent if and only if v is not the zero vector - You can always decide by inspection when a set of vectors in linearly dependent. Row operations are unnecessary. Just check whether at least one of the vectors is a scalar times the other (only applies to sets of two vectors) - A set of 2 vectors is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly dependent if and only if neither of the vectors is a multiple of the other - Geometric terms – two vectors are linearly dependent if and only if they lie on the same line through the o