In this post, we’re going to discover how hard is linear algebra, and what you should do about that.
Is linear algebra hard?
Linear algebra is one of the most difficult courses in math after multivariable calculus in college. It has a lot of vectors and matrices to deal with. In addition requires to have solid basics in algebra courses like algebra 2, and precalculus.
Linear algebra is tough because it is abstract and requires a lot of proofing. In addtion, it requires good spatial memory. students need to have some kind of imagination by representing vectors in multidimensional spaces.
In this article, we’ll discover in the detail the most difficult courses you will be studying in linear algebra to see how hard is linear algebra.
How linear algebra is hard?
linear algebra course is including 3 interesting and the most difficult topics that are:
we’re going to list what are the difficulties that most students face in each of these 3 sections that make linear algebra hard
– In vector and spaces
1 Sometimes it hard to differentiate between notions
The first thing that students find difficulties with is differentiating between these 3 things:
These terms are close to each other but each one is different from the other for instance:
we have a car running with a speed of 15 mph EST.
so in this case, the magnitude is 15 while the direction is EST. But what is a vector in this case?
a vector in this case or velocity called in the combination between speed and direction which is 15 mph EAST.
2 – it is hard to represent some vectors graphically
Another aspect that makes vectors difficult in linear algebra is their representation in space for instance:
or even with a more advanced example :
In order to represent this vector, it will be more difficult than in the previous example. this is just a 3-dimensional space. But it doesn’t stop here.
In linear algebra, you will be forced to represent a vector in multidimensional spaces. so It gets tougher to complete and impossible in multiple dimensions over 4 dimensions.
So the tough thing you will find in vector units is how to present them graphically.
3 – linear combinations
there is another topic that might students find difficult in vectors, called linear combination and span where you should find a combination of all vectors in one area.
what makes this system difficult is it requires resolving system like the example below:
students who have weak basics in algebra systems might find the struggle to solve more difficult systems like these ones:
5 – subspaces
There is another subject in linear algebra which is difficult for some students called subspace, this subject requires having strong photographic memory or imagination. This topic is based on working on a small portion of space.
this part requires having a solid background in algebra especially equations and inequalities and of course arithmetics.
But one of the most difficult things students find in vector linear algebra is solving systems that are complex containing multiple valuables like this example below
In Matrix transformations
1 – functions in matrix transformation
the first difficult subject you will discover in matrix transformation is implementing a function to vectors.
this topic is made to do the linear transformation to vector. In a simple word transform a vector from a given position or group to another group or position.
This operation requires imagination and using matrice to make this transformation. But before that require having some kind of imagination to not make mistakes and have excellent training and experience in matrices.
This example was easier but when we talk about a group of vectors in multidimensional space. It is impossible to do it manually. This is why we use linear transformations in linear algebra.
2 – inverse matrices
One other subject students find difficult is to make the inverse of matrices, it is a little bit complex and requires to be used with matrix transformations.
The other tough topic is to find determinants for matrices especially when we talk about complex matrices like 3×3 or 4×4 or even more. It becomes confusing and requires a lot of hard work to get used to.
3 – Determinants
so this was just a simple example of 3X3 determinants while the linear algebra course covers more in-depth of these determinants. Meaning you will be studying 4X4 determinants and even n dimensions which require specific methods to solve.
The last difficult subject you will be learning in the matrix transformation is transpose of matrix meaning transforming matric columns to rows like this example below.
It will not be going simple like we explained you will be making difficult transpose operations like finding:
In Alternate coordinate systems
In this topic of coordinate systems, you will be focusing a lot on projections. In other words, you will have a lot of projection to do especially on vectors. In addtion, you will need to have absorbed and studied well transposes topics including spaces and subspaces and matrix transformations as we explain at the beginning of the article.
you will also discover a row space that represents a partial row or portion of space. In addtion discover orthonormal bases which abstract math that is hugely intensive in proofs and vectors.
the last topic that studnet strgulle into is eigenvalues and eigenvectors . Eigenvalues are the important numbers associated with a matrix. They are used to calculate how much a matrix is changing.
this topic is tough and requires dealing with a lot of matrix transformations and having solid basics in polynomials.