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$A$,$B$ are two vectors forming a basis in $\mathbb{R}^2$ and $F:\mathbb{R}^2\rightarrow \mathbb{R}^n$ a linear map. Show that either $F(A)$ and $F(B)$ are linearly independent, or the image of $F$ has dimension $1$, or the image of $F$ is $\{0\}$

I have done two of the three cases, I cannot find a way for the second case.

Let $v\in \mathbb{R}^2$ be $aA + b B$ with $a,b$ reals. Then the image of any $v$ is $$F(aA+bB) = aF(A) + bF(b) .$$

Assume $F(A)$ and $F(B)$ are not zero vectors. Then $aF(A) + bF(b)=0$ iff $a=b=0$ as $aA + b B=0$ iff $a=b=0$ and that $F(0)=0$.

If $F(A) = F(b)=0$ then any $v = aA + b B \implies F(v) =0 $ and the image of $F$ would be $\{0\}.$

However, I am unable to prove the third case, i.e its image would have dimension $1$ if not above cases.

This is a problem from Lang's Linear Algebra, which I am trying to learn over the weekend so don't have any other help for trivial doubts.

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3 Answers

up vote 6 down vote accepted

Your reasoning is incorrect.

You are correct that if $a$ and $b$ are zero, then $aF(A)+bF(B)=0$. However, it is not true that if $aF(A)+bF(B)=0$ then you must have $a=b=0$. The error lies in that even though you are right that $aF(A)+bF(B)=F(aA+bB)$, you have no warrant for assuming that just because the image is equal to $0$ then it follows that the vector $aA+bB$ must be zero. This is not true in general.

Instead, notice that the image of $F$ is spanned by $F(A)$ and $F(B)$. Then consider the following two possibilities:

  • Either $F(A)$ and $F(B)$ are linearly independent;
  • or else $F(A)$ and $F(B)$ are linearly dependent.

If $F(A)$ and $F(B)$ are linearly independent, there is nothing to do.

If $F(A)$ and $F(B)$ are linearly dependent, and both are zero, then there is nothing to do: the image is $0$.

And if they are linearly dependent and at least one is nonzero, then either $F(B)$ is a scalar multiple of $F(A)\neq 0$, in which case you should show that the image of $F$ consists of all multiples of $F(A)$ and nothing more (so it is one dimensional); or else $F(A)=0$ and $F(B)\neq 0$, in which case you should show that the image of $F$ consists of all multiples of $F(B)$, and so is one dimensional.

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If $F(A)$ and $F(B)$ are not linearly independent, there are scalars $c_1, c_2$ with $c_1 F(A)+c_2 F(B) = 0$ and $c_1 \ne 0$ or $c_2 \ne 0$, and by switching $A$ and $B$ we may assume without loss of generality that $c_1 \ne 0$.

You can then write $F(A)$ in terms of $F(B)$ (do you see how?) and using that, write an arbitrary linear combination of $F(A)$ and $F(B)$ in terms of just $F(B)$. But if $A$ and $B$ are a basis for $\mathbb{R}^2$, then any vector in $\mathbb{R}^2$ can be written as a linear combination of $A$ and $B$, and the image of that under $F$ will be a linear combination of $F(A)$ and $F(B)$, which by the above is just some scalar multiplied by $F(B)$. The two cases to consider are $F(B)=0$ and $F(B) \ne 0$, the latter of which has dimension 1 image and the former of which has image {0}.

There are quite a few details to be filled in, so hopefully I haven't spoiled it for you.

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SINCE A,B FORM A BASIS OF R^2, THEIR IMAGES SPAN THE RANGE OF F. THAT IS RANGE OF F=SPAN{F(A), F(B)} . CLEARLY DIM RANGE OF F IS EITHER 0,1 OR 2.

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please avoid capital letters –  jim Jan 7 '13 at 10:02
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