In linear algebra(I'm the beginner so far), I feel the concepts and ideas are based on on another, like **layer by layer**, and there are many relations among the properties.

For example, Matrix Multiplication, AB is essentially the composition of Linear Transformation（$A(Bx)=(AB)x$ to manually define AB) , and Linear Transformation is also from Matrix Equation($Ax=b, T(x)=b=Ax$)，and Matrix Equation is also from Vector Equation($x_1a_1+...+x_na_n=b$ equivalent to $Ax=b$)，and finally Vector Equation is the basic System of Linear Equations.

so Vector Equation could be intuitively understood as Linear System，Matrix Equation is just another formation。 Linear Transformation is fine too，but when it comes to Matrix Algebra， like Inverse Matrix and its properties, it seems **very hard** to be intuitively understood through the basic System of Linear Equations. It'll be only fine when I admit the manually definition the matrix multiplication from composition of linear transformation. Because as the process moves on, it's more and more like everything is constructed layer by layer, and more and more complicated as it's very hard to try to understand it from the very **basic layer** (e.g. linear system) **intuitively**.

I'm not sure whether it's the normal situation, because there are also intersected properties among many 'layers'. Therefore the whole 'knowledge structure' is not very clear in my brain.

And, when I think of the the basic Quadratic Formula in elementary algebra, ， its meaning is just to solve the quadratic equation with one variable, and then it is proved from completing square of $ax^2+bx+c=0$ to find general formation of $x$. But it is still impossible to 'understand thoroughly' from the original equation like how the relation among each variable and coefficient. So when I need to use it, I just refer to the book or prove it myself (assume if I forget)

So will it the same thing as learning the Linear Algebra, like the **determinant** of matrix, $\det A = \sum_{j=1}^{n}(-1)^{1+j}a_{1j}\det A_{1j}$. Is it okay that what I need to do is only to know two things,

**What it really is.****How to prove it.**

**(When I use it, just refer to book or prove it, however, prove the determinant takes a little long time).** Because this formula ($\det A$) is already **impossible to be thoroughly understood from the very basic 'layer'** like system of linear equations or even vector equation. I only know, **Firstly** its meaning is to determine whether a matrix is invertible (because invertible matrix must has $n\times n$ size and has pivot position in each row and no zeros in diagonal entries.), **Secondly** it's proved from $n\times n$ matrix row reduced to echelon form, then the right-bottom corner must be nonzero, the above formula is just the brief formation of the item in right-bottom corner.

I know this two things,**could I say I'm already understand it ?** Because even though I know **1.what is really is and 2.how to prove it**, but I'm still not able to remember it when use it.