I have something to discuss about mathematical understanding, and use the Linear Algebra as example.
Sometimes, I feel the intuitive understanding of definitions/theorems itself is very different from understanding its proofs. But for some others especially when it's generalized to abstract level ,or to general n-dimensions. Then I cannot use the 'intuitive' way to understand('see' the picture as 'intuition', for 2/3-spaces), but always the proof is indeed the understanding procedure('see' the algebraic expressions, but 'visible intuition' disappeared.)
Example:
$^t(AB)={^t(B)}{^t(A)}$ , for this one, I cannot 'see' any intuition ,but the proof.$(AB)=\sum\limits_{j=1}^{n}{a_{ij}b_{jk}}=\sum\limits_{j=1}^{n}{b_{jk}a_{ij}}$ , to tranpose it, $^t(AB)=\sum\limits_{j=1}^{n}{b_{kj}a_{ji}}$ is exactly $^t(B){^t(A)}$
Let $A_1, ...,A_r$ be generators of subspace $V$ of $R^n$, and $W$ be set of all vectors in $R^n$ which are perpendicular to $A_1, ..., A_r$, then W is perpendicular to each elments in $V$.
For this, I also cannot 'see' intuition of how it could do that in general space. But:
Proof: Let $w$ be any vector of $W$. Then, $w\cdot t_iA_i=t_i(w\cdot A_i)=0$, for $i = 1,...,r$, then, $w\cdot(\sum\limits_{i=1}^{r}{t_i A_i})=\sum\limits_{i=1}^{r}{t_i(w\cdot A_i)}=\sum\limits_{i=1}^{r}{0}=0$, as we know $\sum\limits_{i=1}^{r}{t_i A_i)}$ is a space generated by $A_i, i=1,...,r$, it concludes our proof.
The only idea I could 'see' is dot product has the distributive law and use the definition of linear combination, it'll be possible to prove its correctness.