Prove that an isomorphism produces a basis of W

**Runty** · September 30th 2010, 08:26 AM

I think I may have the answer for this one, but I could use a second opinion.

Suppose $V$ and $W$ are vector spaces over a field $F$ .

Now suppose $\psi:V\rightarrow W$ is an isomorphism and $\{ v_1,...,v_n\}$ is a basis of $V$ . Prove that $\{\psi(v_1),...,\psi(v_n)\}$ is a basis of $W$ .

Here is what I have so far:

Let $\{w_1,...,w_n\}$ be a basis for $W$ where
$w_1=\psi(v_1),...,w_n=\psi(v_n), \forall n>0$ .
Then we consider the map
$\psi(x)=x_1w_1+x_2w_2+...+x_nw_n$ where $x=x_1v_1+x_2v_2+...+x_nv_n\in V$ .
Since the bases are inverse to each other, we can conclude that $\{\psi(v_1),...,\psi(v_n)\}$ is a basis of $W$ .

I don't know if I have any major holes in my proof, so if you see any, please let me know.

**HappyJoe** · September 30th 2010, 09:39 AM

The last part in particular sounds strange.

What does it mean that the bases are inverse to each other, and why does it imply that $\{\psi(v_1),\ldots,\psi(v_n)\}$ is a basis for $W$ ?

I would do like this: We need to prove that $\{\psi(v_1),\ldots,\psi(v_n)\}$ is a basis for $W$ . So we need to check two things, namely that the vectors in the proposed basis are linearly independent and that they span all of $W$ .

Let $x_1\psi(v_1)+\ldots+x_n\psi(v_n)=0$ for some scalars $x_i$ . Apply linearity of $\psi$ to get

$\psi(x_1v_1+\ldots+x_nv_n)=0.$

Remember that $\psi$ is an isomorphism, so what do you know about vectors that $\psi$ map to 0?

To see that the basis spans all of $W$ , take some vector $w\in W$ , and choose a vector $v\in V$ , such that $\psi(v)=w$ (why does this vector $v$ exist?).

You know that $\{v_1,\ldots,v_n\}$ is a basis for $V$ , so you can write $v$ as a linear combination of these basis vectors. Try applying $\psi$ to this relation and see what happens.

**Runty** · September 30th 2010, 10:23 AM

Originally Posted by HappyJoe

The last part in particular sounds strange.

What does it mean that the bases are inverse to each other, and why does it imply that $\{\psi(v_1),\ldots,\psi(v_n)\}$ is a basis for $W$ ?

I would do like this: We need to prove that $\{\psi(v_1),\ldots,\psi(v_n)\}$ is a basis for $W$ . So we need to check two things, namely that the vectors in the proposed basis are linearly independent and that they span all of $W$ .

Let $x_1\psi(v_1)+\ldots+x_n\psi(v_n)=0$ for some scalars $x_i$ . Apply linearity of $\psi$ to get

$\psi(x_1v_1+\ldots+x_nv_n)=0.$

Remember that $\psi$ is an isomorphism, so what do you know about vectors that $\psi$ map to 0?

To see that the basis spans all of $W$ , take some vector $w\in W$ , and choose a vector $v\in V$ , such that $\psi(v)=w$ (why does this vector $v$ exist?).

You know that $\{v_1,\ldots,v_n\}$ is a basis for $V$ , so you can write $v$ as a linear combination of these basis vectors. Try applying $\psi$ to this relation and see what happens.

Thanks a lot for the help. That "bases are inverse to each other" part, I actually got off Wikipedia. It applied to ordered bases, though, so that could've made a difference.

Either way, I've gotten it now so, again, thanks a bunch.

**HallsofIvy** · October 1st 2010, 04:05 AM

Originally Posted by Runty

I think I may have the answer for this one, but I could use a second opinion.

Suppose $V$ and $W$ are vector spaces over a field $F$ .

Now suppose $\psi:V\rightarrow W$ is an isomorphism and $\{ v_1,...,v_n\}$ is a basis of $V$ . Prove that $\{\psi(v_1),...,\psi(v_n)\}$ is a basis of $W$ .

Here is what I have so far:

Let $\{w_1,...,w_n\}$ be a basis for $W$ where
$w_1=\psi(v_1),...,w_n=\psi(v_n), \forall n>0$ .

You appear to be assuming what you want to prove! That is that $w_1=\psi(v_1),...,w_n=\psi(v_n)$ is a basis.

Instead you need to show that this set (1) is independent and (2) spans the space.

To show independence: Supppose that [tex]a\psi(v_1)+ b\psi(v_2)+ \cdot\cdot\cdot+ z\psi(v_n)= 0[/itex]
Since $\psi$ is a linear transformation, that is the same as $\psi(av_1+ bv_2+ \cdot\cdot\cdot+ zv_n)= 0$

Since $\psi$ is an isomorphism, its kernel is the 0 vector only. Therefore $av_1+ bv_2+ \cdot\cdot\cdot+ av_n= 0$ . What does that tell you?

To show that $\{\psi(v_1), \psi(v_n), \cdot\cdot\cdot, \pis(v_n)\}$ span W, let w be any vector in W. Since $\psi$ is an isomorphism, there exist v in V such that $\psi(v)= w$ . Since [tex]\{v_1, v_2, \cdot\cdot\cdot, v_n} is a basis for V, there exist scalars a, b, ..., z such that $av_1+ bv_2+ \cdot\cdot\cdot+ zv_n= v$ . Apply $\psi$ to both sides of that equation.

Then we consider the map
$\psi(x)=x_1w_1+x_2w_2+...+x_nw_n$ where $x=x_1v_1+x_2v_2+...+x_nv_n\in V$ .
Since the bases are inverse to each other, we can conclude that $\{\psi(v_1),...,\psi(v_n)\}$ is a basis of $W$ .

I don't know if I have any major holes in my proof, so if you see any, please let me know.

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