Algebra Vs Analysis
I've come to like linear algebra far more than analysis and was wondering how much analysis one would be required to learn in order to pursue a career in research in an algebraic field. For example, at the graduate level, if I were interested in focusing on algebraic topology or algebraic geometry then would learning the contents of analysis 2, real analysis, and differential equations be enough before one exclusively focuses on algebra core subjects? Is it common for mathematicians of one field to be very weak in other fields? Like somebody who does exceptional research in algebraic geometry who would struggle with solving problems in a complex analysis text?
Thanks in advance!
There is no good answer about what is "enough". But I can say two things:
Making a "specialization decision" very early is definitely a bad idea. Basically you are deciding that you don't like/don't care about/don't need topics you know nothing about.
Even if you make the decision to specialize, you cannot predict years in advance what tools or ideas might be useful to you. Many of the most significant advances in math, either in a global sense or in a personal sense (as in one's own research) are often fueled by knowledge of other areas. The more math you know, the more ideas available to you.
Final thought: every top mathematician I ever met had a broad knowledge, way beyond their specialization.
There are some subjects that are required and occasionally mandatory to study in a graduate program. A graduate level course in Complex Analysis, Real Analysis and Algebraic Topology, Abstract Algebra are usually mandatory. Before focusing on a specific area, even if you decide about this very early, you need to cover certain graduate-level coursework in any case.
Linear algebra appears in many areas e.g. Differential Geometry, and is a subject required to know well enough for pursuing graduate studies. So enjoying this subject shouldn't imply that you should pursue one area or another.
That is why before starting research or focusing on a certain area you either pursue a master's degree to cover material in certain areas and then enter a PhD program or go through a 1-2 years preparation within a graduate program in order to learn and finalise your decision.
At the graduate level you'll have to know standard (advanced) textbook material in both algebra and analysis to pass your qualifying exams. You shouldn't have to struggle too hard with the problems.
When you start focusing on your own research your abilities in the unrelated fields will get rusty. Years later the then current graduate curriculum might give you trouble unless you went back to study it anew.
It's always a good idea to keep somewhat current in several areas. Some of the major outstanding problems (e.g. the Langland's program ) are precisely about how different fields shed light on one another.
The other answers have made all the important general points, IMO, so I will add a couple of things from my own experience.
Like you, my tastes favor the algebraic end of the spectrum, plus (geometric and algebraic) topology; I never much cared for real analysis.
When I encountered complex analysis, it was a revelation: this stuff is beautiful. Totally different feel from real analysis.
Functional analysis is a blend of linear algebra and analysis, and while I don't rank it quite as high in my own personal mathematical beauty contest, I would be sorry to have missed out on it.
The moral has already been stated by Argerami: "Basically you are deciding that you don't like/don't care about/don't need topics you know nothing about."
Finally, since you like linear algebra, take a look at Hoffman & Kunze sec. 6.4 for a nice application of it to the theory of ordinary differential equations.