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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square.

a) If two rows of \(A\) are the same, show that \(A\) is not onto by finding a vector \(y=\) \(\left(y_{1}, \ldots, y_{k}\right)\) that is not in the image of \(A\). [HINT: This is a mental computation if you write out the equations \(A x=y\) explicitly.]
b) What if \(A: \mathbb{C}^{n} \rightarrow \mathbb{C}^{k}\) is a complex matrix?
c) More generally, if the rows of \(A\) are linearly dependent, show that it is not onto.
in Mathematics by Platinum (129,882 points) | 154 views

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