For a nonzero matrix in row echelon form, is it always true that the non zero rows are always linearly independent?
I thought for a long time but still couldnt come out with an answer. Im just too weak in my concepts. Can anyone help clarfy?
[Math] For a nonzero matrix in row echelon form, is it always true that the non zero rows are always linearly independent
linear algebra
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Best Answer
Yes. $$ \begin{pmatrix} 1 & M_{12} & M_{13} & \cdots & M_{1m} & \cdots & M_{1n}\\ 0 & 1 & M_{23} & \cdots & M_{2m}& \cdots & M_{2n}\\ \vdots & & \ddots & & & & \vdots\\ 0 & 0 & 0 & \cdots & 1 & \cdots & M_{mn}\\ \end{pmatrix} $$
Write $v_1$ for the first row, $v_2$ for the second row, etc. Suppose $$ a_1v_1 + \cdots + a_nv_n = 0. $$ We have to prove that each $a_i = 0$. Well, the expression is just $$ \begin{pmatrix} a_1 & M_{12}a_1 + a_2 & M_{13}a_1 + M_{23}a_2 + a_3 & \ldots \end{pmatrix} $$ the $i$th entry in this vector looks like $$ a_i + \text{stuff that is 0 if we know that } a_j \text{ is } 0 \text{ for } j < i $$ Working left to right, we see that $a_1$ must be 0; therefore $a_2$ must be zero, therefore $\ldots$