From: 3blue1brown
Linear algebra is a foundational subject required for almost any technical discipline, yet it is often poorly understood by students taking it for the first time [00:00:16]. While students may learn to compute various operations such as matrix multiplication, determinants, cross products, or eigenvalues, they often lack an understanding of the underlying “why” behind these operations or what certain concepts like an eigenvalue truly represent [00:00:30].
The Disconnect: Computation vs. Intuition
Many students become well-practiced in the numerical operations of matrices but remain only vaguely aware of the geometric intuitions that underpin them [00:00:53]. There is a fundamental distinction between understanding linear algebra on a numeric level and grasping it on a geometric level [00:01:01].
- Geometric Understanding The geometric understanding allows one to judge which tools to apply for specific problems, comprehend why they work, and correctly interpret the results [00:01:08].
- Numeric Understanding The numeric understanding is what enables the actual execution of those tools [00:01:19].
While both are important, the geometric perspective is crucial for deeper comprehension and effective problem-solving [00:01:08].
Consequences of Lacking Geometric Understanding
Without a solid foundation in the geometric understanding of linear algebra, problems can go unnoticed until a student delves deeper into their chosen field, whether it be computer science, engineering, statistics, economics, or even mathematics itself [00:01:25]. In environments that assume fluency with linear algebra, the application of the field by professors or co-workers might appear to be “utter magic” [00:01:45]. They quickly identify the correct tools and anticipate approximate answers, which can seem like “computational wizardry” if one assumes they are performing all calculations mentally [00:01:52].
Analogy: The Sine Function
Consider learning the sine function in trigonometry primarily through its infinite polynomial expansion, which is how calculators compute it [00:02:04]. If the geometric relation to triangles was vague or not emphasized, later physics courses where sines and cosines are readily applied would be intimidating, making it seem like only those with “computers for brains” are suited for the subject [00:02:34].
Bridging the Gap
Fortunately, much like trigonometry, linear algebra is underpinned by a handful of visual intuitions [00:03:02]. The connection between computation and these visual intuitions is typically straightforward in linear algebra [00:03:07]. By internalizing these geometric intuitions and understanding their relationship with the numerical aspects, the subject’s details and its practical applications become much clearer [00:03:14].
Although most professors strive to convey geometric understanding, many courses still allocate a disproportionate amount of time to the numerical side [00:03:25]. In modern practice, computers handle the numerical computations, while humans focus on the conceptual understanding [00:03:38].
A series of videos aims to animate these intuitions, starting from the basics of vectors and progressing through the core topics of linear algebra [00:03:48]. The goal is not to teach the entire subject, but to lay down the correct intuitions to make future learning more productive and fruitful [00:04:04]. This resource can also assist educators in directing students who need a quick refresher [00:04:21].