Linear algebra – Which statements are correct

linear algebraorthogonalityproductsvector-spaces

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.

Best Answer

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

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

Related Question