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mathlife
4. Let A ∈ F^m×n . Show that if y ∈ F^1×n is in the row space of A and x ∈ F^n×1 is in the null space of A, then yx = 0
Well, if \(x\in \ker(A)\) (\(\ker(A)\) is the null space), then \(Ax=0\). Let \(A=(\vec{a_1},\vec{a_2},...,\vec{a_n})\) where each \(\vec{a_n}\) is the \(n\)-th row vector. Since \(y\) is in the row space of A, we can say that \[y=r_1\vec{a_1}+...+r_n\vec{a_n}\]for some \(r_1,...,r_n\in F\).Then. \[yx=r1\vec{a_1}x+...+r_n\vec{a_n}x.\]Since \(Ax=0\), \(\vec{a_j}x=0\) for all \(1\le j\le n\), we can conclude that \(yx=0\). Did this make sense?