Linear algebra - Which statements are correct?

Question

Question: Which statements are correct?

1. If $\|u\|^2 + \|v\|^2 = \|u + v\|^2$, then $u$ and $v$ are orthogonal.
2. For any scalar $c$, $\|cv\| = c\|v\|$
3. If $x$ is orthogonal to every vector in subspace $W$, then $x$ is in $W^\perp$
4. For an $m\times n$ matrix $A$, vectors in the null space of $A$ are orthogonal to vectors in the row space of $A$
5. $u \cdot v - v \cdot u = 0$

I think each of them is correct because for
1. Pythogorean Theorem
2. $\|cv\| = \sqrt{c^{2} v_1^{2} + c^{2} v_2^{2} + \cdots} = \sqrt{c^2}\cdot\|v\| = c\|v\|$
3. N.A.
4. $\operatorname{Row}(A)^\perp = \operatorname{Nul}(A)$
5. $u \cdot v = v \cdot u$

but this is wrong, i.e. at least one of them is incorrect. Which one(s) is it?

Thanks.

Answer 1

For the first one we can use that

$$\|u+v\|^2=(u+v)\cdot(u+v)=\|u\|^2+\|v\|^2+2u\cdot v$$

For the second one it should be

$$\|cv\|=|c| \cdot \|v\|$$

Answer 2

1. The equation is equivalent to $\langle u|v\rangle=-\langle v|u\rangle$, so is equivalent to orthgonality in real spaces but not complex ones, where it only requires $\langle u|v\rangle$ to be imaginary.
2. It should be $\Vert cv\Vert=|c|\Vert v\Vert$.
3. That's the definition of $W^\perp$.
4. $v$ is in $A$'s right null space (aka kernel) iff $\sum_jA_{ij}v_j=0$, while $w$ is in $A$'s row space iff constants $c_i$ exist with $w_j=\sum_ic_iA_{ij}$. Then $\langle v|w\rangle=\sum_jv_jw_j=\sum_{ij}c_iA_{ij}v_j=0$, so the claim is true. Similar, vectors in $A$'s left null space (aka cokernel) are orthogonal to $A$'s column vectors.
5. As per 1.