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Let $V$ be a real finite-dimensional vector space and $f$ and $g$ non-zero linear functionals on $V$. Assume that $\ker(f)$ is a subset of $\ker(g)$. Pick out the true statements.

  1. $\ker(f) = \ker(g)$.

  2. $\ker(g)/\ker(f) = \mathbb{R}^k$ for some $k$ such that $1\leq k < n$.

  3. There exists a nonzero constant $c$ such that $g = c$.

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Hint: $\mathrm{dim}(\mathbb{R}^n/\mathrm{ker}(f))\leq 1$ for all $f\in V^*$ – Norbert Aug 7 '12 at 11:40
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@DavideGiraudo How can you distinguish this as a typo from a purposefully true/false case included by the original author of the question? I agree with you we need to get the OP to doublecheck, though :) – rschwieb Aug 7 '12 at 11:59
@rschwieb it is assumed that $g$ is linear, and so it cannot be equal to a nonzero constant. I suspect it is more likely to be a typo than a truely false (!) statement – M Turgeon Aug 7 '12 at 13:09
@MTurgeon Well anyone can impose their expectations upon the points, but I mean logically speaking there is no way to detect the difference between accidentally false and purposefully false points. Well, other than asking the OP to make sure it's really what s/he meant. – rschwieb Aug 7 '12 at 13:23
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Ranadip, this is the 9th time you have asked a question which looks like homework copied straight from a text with no motivation, no mention of what you have tried, and without the homework tag, all of which are required. This is not acceptable. If you cannot follow this community's norms then you should not post here. – Nate Eldredge Aug 7 '12 at 13:50
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closed as not a real question by BenjaLim, Did, Leonid Kovalev, Asaf Karagila, J. M. Aug 17 '12 at 0:34

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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A very useful hint in this case (which is a variant of the one Norbert has given in the comments) is to use the Rank-Nullity theorem. This allows you to draw conclusions about the dimensions of the kernels.

That sort of resolves 1 and 2 at the same time.

For #3, perhaps recall what a matrix representation should look like for a map from $\mathbb{R}^k$ into $\mathbb{R}$.

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