Given a vector $x \in \mathbb{R}^n$ we know the following inequality holds for the product of the vector $x$ and a matrix $A \in \mathbb{R}^{m\times n}$ i.e., $Ax=y$ where $y \in \mathbb{R}^m$
$\Vert y\Vert_2=\Vert Ax\Vert_2\leq \Vert A\Vert_F\Vert x\Vert_2$
1) Can we say $x$ is linearly independent of rows of $A$ when inequality (<) holds.
or in other words
1) Can we say $x$ is linearly dependent on rows of $A$ when equality (=) holds.
Best Answer
We have $=$ if and only if $x$ and each row of $A$ are linearly dependent. This follows directly from Cauchy Schwarz inequality.
Proof: Let $a_i^T$ denote the $i$-th row of $A$. Then, we have $$ \| Ax \|^2 = \sum_{i=1}^m (a_i^T x)^2 \le \sum_{i=1}^m \|a_i\|^2 \|x\|^2 = \|A\|_{\mathrm F}^2 \| x \|^2, $$ where the inequality follows from Cauchy Schwarz. Now, by Cauchy Schwarz, we have $(a_i^T x)^2 = \|a_i\|^2 \|x\|^2$ if and only if $a_i$ and $x$ are linearly dependent.