Attempt to proof following statement (kernel and image of a matrix)

Question

Let $\{\overrightarrow u, \overrightarrow v,\overrightarrow w\}$ be an orthonormal basis of $\mathbb R^3$ with respect to the standard inproduct. We define the matrix $A$ as $A= \overrightarrow u\overrightarrow u^{tr} + \overrightarrow v\overrightarrow v^{tr} + \overrightarrow w\overrightarrow w^{tr}$.

Then show that $A=I_3$.

Hint: Use the preceding exercise and show that $Ker(A-I_3)= \mathbb R^3$.

(i) Let $\overrightarrow u = \{\overrightarrow {u_1},\overrightarrow {u_2}, \overrightarrow {u_3} \}, v=\{\overrightarrow {v_1},\overrightarrow {v_2},\overrightarrow{v_3}\},w=\{\overrightarrow {w_1},\overrightarrow {w_2},\overrightarrow {w_3}\}$ be orthonormal bases (That means you can treat them like $\{\overrightarrow {e_1}, \overrightarrow {e_2}, \overrightarrow {e_3}\}$ of $\mathbb R^3$). This means the value on the diagonals will be equal to $1$ in a matrix of $3\times 3$.

View it as

$A= \begin{bmatrix} \overrightarrow {u_1} & \overrightarrow {v_1} & \overrightarrow {w_1} \\ \overrightarrow {u_2} & \overrightarrow {v_2} & \overrightarrow {w_2} \\ \overrightarrow {u_3} & \overrightarrow {v_3} & \overrightarrow {w_3} \end{bmatrix}\cdot \begin{bmatrix} \overrightarrow {u_1^{tr}} & \overrightarrow {v_1^{tr}} & \overrightarrow {w_1^{tr}} \\ \overrightarrow {u_2^{tr}} & \overrightarrow {v_2^{tr}} & \overrightarrow {w_2^{tr}} \\ \overrightarrow {u_3^{tr}} & \overrightarrow {v_3^{tr}} & \overrightarrow {w_3^{tr}} \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0& 0& 1 \end{bmatrix}= I_3$

(ii) Well if $A= I_3$ then we can conclude from the previous excercise that it is equal to $\mathbb R^3$.

L.S.:

Will proofs like this (i.e. informal proofs) if completed be accepted in maths journals or if someone claims to have a solution on one or more of the remaining 6 Millenium Prize Problems?

Answer 1

The fact that you are treating $u,v,w$ like $e_1,e_2,e_3$ of $\mathbb R^3$,means nothing at all. I do not see any similarity between $u,v,w$ an the standard basis, apart from the fact that both are orthonormal.

The way to see $A$, which you have written, is incorrect. In fact, if say $u = (u_1,u_2,u_3)$, then $uu^T$ is a $3 \times 3$ matrix, whose $(i,j)$th entry is $u_iu_j$. Similarly with $v$ and $w$. So, what you have is: $$ A_{ij} = u_iu_j + v_iv_j + w_iw_j \quad 1 \leq i,j \leq 3 $$ You can see, for example $A_{11}$ does not match the description I gave.

For example, $A_{23} = u_2u_3 + v_2v_3 + w_2w_3$. It is not at all clear why this entry should be zero.

---

To show that $A = I_3$, we use the exercise given : show that $\ker(A-I_3) = \mathbb R^3$. I will now only say $I$ rather than $I_3$.

Naturally, we must first find some elements in the kernel. Let's try operating $A - I$ on $u$. What is $Au - Iu$? $$ Au - Iu = uu^Tu + v(v^Tu) + w(w^Tu) - u = u +0+0 - u = 0 $$

so $(A-I)u = 0$. Note that $v^Tu = w^Tu = 0$ from orthogonality, and $u^Tu = 1$ from normality.

• Show that $(A-I)v = (A-I)w = 0$ similarly.

• So the kernel of $(A-I)$ contains $u,v,w$, which are three linearly independent elements of $\mathbb R^3$. Can you see why the kernel must be equal to $\mathbb R^3$?

• Now use the exercise to complete the problem.

Answer 2

Short answer: no, informal proofs are not permitted, at least, not this informal.

There will always be steps skipped in a mathematics research paper; nobody wants to read every step. However, the skipped steps should be within easy grasp of the reader, able to be filled in with relatively little effort for somebody in the field.

But, importantly, those steps have to exist. There has to be a way to fill in those gaps, all of them, right from axioms to the theorem proven. Maybe some are in a theorem printed elsewhere, or maybe they are just skipped for expediency, but they should be accessible to the interested reader.

This is the problem with your proofs. There are a number of issues, but I think this statement is fairly representative:

Let $\overrightarrow u = \{\overrightarrow {u_1},\overrightarrow {u_2}, \overrightarrow {u_3} \}, v=\{\overrightarrow {v_1},\overrightarrow {v_2},\overrightarrow{v_3}\},w=\{\overrightarrow {w_1},\overrightarrow {w_2},\overrightarrow {w_3}\}$ be orthonormal bases (That means you can treat them like $\{\overrightarrow {e_1}, \overrightarrow {e_2}, \overrightarrow {e_3}\}$ of $\mathbb R^3$).

I understand that you're going for an informal, more intuitive tone here, but it's not clear what's under the intuition. What exactly does it mean to "treat" one basis like another? Do they produce the same coordinate vectors? Since these bases can be treated as the same basis, does this mean we don't have to use any change of basis formulas? In what circumstances can we replace our bases with the standard basis?

By not being properly clear, you've left the connection between your previous statement and your next statement nebulous. Basically, you've then leapt to writing down the conclusion, in full. Somewhere in that gap of similar "treatment", there needs to be substantial logic, and it's not clear that it occurs in your proof.

Of course, learning how to write proofs properly is not easy. There's a lot of really tricky little issues and conventions that you need to learn to navigate. This proof is nowhere near a research standard, but nobody writes research standard proofs out of the gate.