One approach for dealing with this problem: make a three-part worksheet. Part one is "stuff using the formulas we just learned today." Part two is "review of old stuff." Part three is "mixed problems."
In essence, when you teach the first concept, the student isn't learning it as an if-then statement: "when you see this, then do this." The student is just learning "do this," because that's the only sort of problem the student sees. So when you introduce the second concept, really the student has twice as many new things to learn - how to solve the second type of problem, and how to distinguish the first type of problem from the second type of problem. Of course there's no getting around this. You have to introduce something first. But that's my theory for why the second thing you introduce makes things slower.
Mathematics is really about relations between things. Therefore, while constructions are good and useful, you should never take them very seriously.
Construction agnosticism
Consider the real and complex numbers. Given $\mathbb R$, you can construct a ring which is isomorphic to $\mathbb C$ by taking pairs of real numbers and defining addition and multiplication in the usual way. Note here that I say that you can define a ring isomorphic to $\mathbb C$. We can ask the following question:
- Is the ring we have defined actually the ring of complex numbers $\mathbb C$ itself?
But you shouldn't ask yourself that question. (Just because you can ask a question, doesn't mean you should.) It won't do you any harm; it's just not useful to.
What really matters? That there is a ring which has all the properties that we want from the complex numbers — such a ring exists. Then we can say: let $\mathbb C$ be such a ring, and then study the maps $f: \mathbb C \to \mathbb C$.
When we say "the" complex number field, the word "the" is a red herring; we just want to talk about some ring which works that way. Similarly, we don't really care about "the" real numbers. If $\mathbb C$ is "the" complex number field, then it contains a principal ideal domain $Z$ coinciding with the abelian group generated by the multiplicative identity; a quotient field $Q$ induced by $Z$; and a subfield $F$ which is the analytic completion of $Q$. We can then call these "the" real numbers. Just as we don't care if "the" complex numbers consist of ordered pairs of objects from some ring $S$, we don't care if "the" real numbers are the ring $S$ or some set of ordered pairs $(s,0)$ for $s \in S$ and $0$ the additive identity of $S$. It just doesn't matter — we just care about the proper subfield $F \subseteq \mathbb C$ which has all of the same properties as the real numbers, so we may as well adopt the convention that this subfield is the field of real numbers.
This applies to the integers as well. The von Neumann construction of the natural numbers makes $3 = \{ \varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\}\} $. Does this mean that $3$ "really is" a set which e.g. has the empty set as a member? Not really, because these ideas are totally irrelevant to what we care about the number $3$. We could consider any other "construction" of the natural numbers, in which case $3$ might not be a set at all (for instance, if we consider a set theory in which the natural numbers are atoms), in which case it is not only irrelevant to consider the maps $f:3\to3$, but these would not even be defined. All we care about is that $3$ is part of a collection of objects $\mathbb N$ which forms a monoid with some specific properties. The "true identity" of $3$ is beside the point.
Mathematical "interfaces" (in place of "foundations")
If this ambiguity bothers you, you can think of it axiomatically as follows: treat each of the sets we care about — such as $\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ — as underspecified objects, where we specify all of the properties about them that we could care about, and only those properties.
von Neumann's construction of ordinals describes a certain countable well-ordered monoid: we don't define $\mathbb N$ to be that monoid, but merely say that it is isomorphic to it, leaving further details to be filled in later.
From equivalence classes of ordered pairs of elements of $\mathbb N$, you can define an ordered ring $Z$, which contains a monoid $M \cong \mathbb N$ and which is the closure of $M$ under differences. Now, $\mathbb N$ can never be a subset of this set of equivalence classes; but it can be a subset of some other set. We never pretended to characterize precisely what object $\mathbb
N$ is, so who is to say that $\mathbb N$ is not itself contained in
a ring which is isomorphic to $Z$? Nobody, that's who; without loss of generality we
may define $\mathbb Z$ to be a ring isomorphic to $Z$, and declare as
a refinement of the earlier specification that in fact $\mathbb N$ is
contained in $\mathbb Z$.
- We may similarly declare that $\mathbb Z \subseteq \mathbb Q$, where $\mathbb Q$ is isomorphic to the ring of equivalence classes of ordered pairs over $\mathbb Z$ in the usual way. We also declare that $\mathbb Q \subseteq \mathbb R$, where $\mathbb R$ is isomorphic to the field of Dedekind cuts over $\mathbb Q$, or (equivalently) to the field of equivalence classes of Cauchy sequences, or any of the typical constructions of the real numbers. The set $\mathbb R$ isn't defined to be any particular one of these constructions, because (a) any of these constructions is as good as the others, and (b) we don't really care about any of the details lying underneath any of the constructions, so long as the properties we care about hold for each.
You should think of these refinements as axioms which we add during the process of doing mathematics.
A definition is in the first place is only an axiom: one which defines a constant, such as defining ∅ by asserting ∀x:¬(x∈∅). These mathematical underspecifications — mathematical interfaces — are also axioms: having proven that a certain sort of monoid exists satisfying the Peano axioms, we assert that $\mathbb N$ is such a monoid, saying nothing more until it suits us to; and similarly declaring that $\mathbb C$ is a sort of number field of a kind which we've proven exists, and which happens to contain the field $\mathbb R$ which we mentioned previously without quite defining it completely.
Fundamentally, this approach to mathematics is not really all that different from what we usually do: it merely substitutes complete descriptions (what Bertrand Russell would call simply "a description") for objects, with partial descriptions. But pragmatically, in the real world as in mathematics, partial descriptions tend to be all that we care about (and in the real world, they are all that we ever have access to). Embracing this allows you to focus on what really matters.
If you are a mathematical "realist", in which the real numbers has an identity separate from our descriptions of it and has some fixed location in the mathematical firmament, this sort of wishy-washiness as to the "exact identity" of these objects may bother you. After all, if you imagine the possible identities of the objects $\mathbb N$, $\mathbb Z$, $\mathbb R$, etc. as you subsume them into more and more complicated objects, it would seem that the set of objects with which a set such as $\mathbb N$ could be identified recedes to infinity as our mathematical framework grows more elaborate. To this I can only say, "so much the worse for realism". If you want the freedom to construct objects and only concern yourself with the relationships that matter, in the end it is better to abandon this preoccupation with the precise identity of a mathematical object, and engage in mathematics as the creative, descriptive, and above all incomplete and ongoing endeavor that it is.
Best Answer
Here are some suggestions that may help.
$\bullet$ How to self study Linear Algebra. See my response and the other responses.
Study Habits from Math Stack Exchange
$\bullet$ What are some good math study habits?
$\bullet$ How to study math to really understand it and have a healthy lifestyle with free time?
$\bullet$ How can I learn to read maths at a University level?
Free Resources
$\bullet$ MIT Open Course Ware. You can also find others at other universities and these have free course notes and video lectures.
$\bullet$ Khan Academy.
$\bullet$ Wiki Books Linear Algebra Resources
$\bullet$ Linear Algebra for Communications: A gentle introduction. I would study this to give you context!
$\bullet$ College Libraries: Peruse the books in the library and see if any fit the style you like.
Book Recommendations
$\bullet$ Linear Algebra Through Geometry, Thomas Banchoff, John Wermer (since you are visual)
$\bullet$ Practical Linear Algebra: A Geometry Toolbox, Gerald Farin, Dianne Hansford (since you are visual)
$\bullet$ Linear Algebra Done Right, Sheldon Axler
$\bullet$ Linear Algebra and Its Applications, David C. Lay
$\bullet$ Linear Algebra Textbook Recommendations on MSE
$\bullet$ Alternative to Axler's “Linear Algebra Done Right”
Problem Books
It is very helpful to do problems and practice, practice, practice to reinforce concepts!
$\bullet$ Linear Algebra Problem Book, Paul R. Halmos
$\bullet$ The Linear Algebra a Beginning Graduate Student Ought to Know, Jonathan Golan.
$\bullet$ 3,000 Solved Problems in Linear Algebra, Seymour Lipschutz
$\bullet$ Schaum's Outline of Linear Algebra Fourth Edition, Seymour Lipschutz, Marc Lipson
History of Linear Algebra
Linear Algebra History Websites
Regards