Ask Question
1 March, 22:19

Let A be an m x n matrix. Show that if A has linearly independent column vectors, then N (A) = {0}.

+5
Answers (1)
  1. 1 March, 22:33
    0
    Let the column vectors of A be: v1, v2, ..., vn If the matrix A has linearly independent column vectors, then the only linear combination of the vectors that equals 0 is the zero combination. In other words, if: c1 v1 + c2 v2 + ⋯ + Cn. Vn = 0⟹ c1 = c2 = ⋯ = Cn = 0 However: c1 v1 + c2 v2 + ⋯ + Cn. vn = 0⟺Ax=0 where x is the vector x = (C1, C2, ..., Cn) and if Ax=0, this means that x is in the null space of A. Since the only vector that makes that linear combination 0 is the 0 vector, it follows that the 0 vector is the only vector in the null space.
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “Let A be an m x n matrix. Show that if A has linearly independent column vectors, then N (A) = {0}. ...” in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers