Linear algebra is a key area of mathematics, especially for those interested in machine learning. While it is essential to learn, it can also be difficult. This article explores why linear algebra can be tough and offers tips to make it easier to understand.

Linear algebra is filled with complex concepts and theories. Despite this, the tools and notations it provides are crucial for those working in fields like machine learning. Because of its importance, linear algebra is often referred to as the "mathematics of data."

At its core, linear algebra focuses on linear combinations. This involves performing arithmetic operations on columns and arrays of numbers, known as "vectors" and "matrices," to create new sets of numbers.

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You will encounter topics such as lines, planes, vector spaces, and mappings, all of which are necessary for understanding linear transformations.

A Closer Look at Linear Algebra

Linear algebra is a relatively recent branch of mathematics, formalized in the 1800s as a way to explore linear equations. If you're studying advanced linear algebra, it's helpful to refresh your knowledge of linear equations.

What is a Linear Equation?

A linear equation consists of several mathematical terms and operations, with some terms being unknown. Here's an example of a linear equation:

Linear equations represent lines on a two-dimensional plane. These lines are created by substituting random values for the unknown "x," which in turn determines how the equation impacts the value of "y."

You can create a whole system of linear equations following this same format, whether it involves one unknown, two, or even more. Here's an example to illustrate:

In linear algebra, "data columns" refer to columns that contain the coefficients and are represented as matrix A.

The unknowns, or variables, are organized into a vector, X, that you need to solve for.

The outputs are shown as a column vector, B.

To simplify the expression, linear algebra notation can be written like this:

The unknowns or variables can be calculated as:

Although many people find linear equations manageable, linear algebra introduces more complexity.

Is Linear Algebra a Difficult Subject?

Many students find linear algebra a tough subject to tackle. Compared to discrete mathematics, which is typically taught in the first year of most STEM programs, linear algebra is more challenging. It is usually introduced in the second year of studies and requires strong analytical and reasoning skills.

Linear algebra is often considered an "intermediate-level" math course, similar in difficulty to calculus II. However, more advanced courses like topology and abstract algebra are even more complex.

Before diving into linear algebra, it’s helpful to have a solid understanding of calculus I, as it forms the foundation for many higher-level math courses.

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While linear algebra can seem complex, it’s manageable with consistent practice. Working through problem sets, practicing regularly, and thinking through equations logically will deepen your understanding. Reading additional textbooks on the subject can also provide clarity and help improve your skills over time.

What Makes Linear Algebra Challenging?

You may be wondering what makes linear algebra so tough. Simply put, the subject is not very intuitive. It places a heavy emphasis on rigorous proofs, and its abstract nature can make it hard to visualize. Unlike other areas of math, you can’t easily picture a linear algebra equation on a graph in your mind, and you don’t see it fluctuate to give different results. This lack of clear visualization can make the concepts harder to grasp and practice.

Another challenge is that linear algebra is different from the math you may have encountered in high school or earlier college courses. In those courses, students often visualize mathematical functions graphically. However, in linear algebra, you’ll be working more with algebraic manipulations, which move away from visual forms. Adjusting to this shift in thinking can be tough, but it’s necessary to understand the algebraic material at the core of the subject.

Some specific areas that students often find challenging include defining mathematical structures using axioms, understanding eigenvectors, and grasping concepts like abstract vector spaces and linear independence.

The difficulty in linear algebra comes from the need to first understand key terms and definitions. Once you’ve done that, the next step is figuring out which calculations and methods to apply in order to reach the correct answer. This process can be complex for many students, adding to the overall difficulty of the subject.

Key Elements that Add to the Difficulty

Linear algebra includes a number of abstract concepts that are essential to its study. For example, you need to focus first on calculations before diving into the more abstract ideas and definitions. This can feel overwhelming for many students, but it’s important to work through these steps systematically.

Another tricky aspect is writing proofs. While the initial concepts may seem straightforward, the material becomes increasingly difficult as you progress. The basics of linear algebra don’t typically involve calculus, but as you dive deeper into mathematics, you’ll find that linear algebra often overlaps with calculus, especially in advanced topics.

The intersection of linear algebra with other fields, like physics, economics, and statistics, also adds to the complexity. As you encounter linear algebraic problems, you'll need a deeper understanding of concepts from other areas of math.

Although linear algebra is sometimes compared to calculus in terms of difficulty, it is generally easier than basic calculus. However, excelling in calculus can still be difficult, even with a strong grasp of theorems. In linear algebra, once you understand the key theorems, solving problems becomes more straightforward.

Adopt an Intuitive Approach to Linear Algebra

Often, we struggle with concepts because we see them as too difficult and don’t take the time to fully understand them. Linear algebra is no different. Instead of feeling frustrated or overwhelmed, try to break down the subject and view it intuitively.

Once you do that, you'll find that linear algebra can actually be interesting and manageable.

Organize Your Operations and Inputs

One of the most challenging aspects of linear algebra is working with matrices. However, the task becomes much simpler when you learn how to organize your inputs and operations. Here's how you can think about it:

1. You have a set of inputs you need to keep track of.

2. You perform predictable linear operations on these inputs.

3. You end up with an output, which might be transformed again.

To begin, focus on tracking the inputs. One simple way to do this is to list them:

• x

• y

• z

Or, you could write them as a vector: (x, y, z).

Next, focus on the operations. In linear algebra, many operations involve basic arithmetic, such as multiplication by a constant and addition. For example, if the operation “F” is defined as:

F (x, y, z) = 2x + 3y + 4z

You can represent this operation as the vector (2, 3, 4). This shows that you will multiply the first input by 2, the second input by 3, and the third input by 4, then add the results together.

If you only consider the first input, you would have:

G (x, y, z) = 2x + 0y + z = (2, 0, 1)

Now, let’s make it more interesting by considering how to handle multiple sets of inputs at once. If you want to run operation F on two sets of inputs, like (a, b, c) and (x, y, z), you might be tempted to write:

F (a, b, c, x, y, z)

But this doesn’t work because F is set up to handle only three inputs at a time, not six. The solution is to separate the inputs into groups:

• 1st Input: a, x

• 2nd Input: b, y

• 3rd Input: c, z

This way, your inputs are organized neatly.

What if you need to apply multiple operations to the same inputs? It’s helpful to set up a row for each operation. For example:

• F: 2 3 4

• G: 2 0 1

Now you’ve organized your inputs in vertical columns, and the operations are in horizontal rows. This method makes everything clearer and more manageable.

Picture the Matrix

To understand linear algebra better, it's helpful to visualize a matrix. This involves imagining the inputs, operations, and the resulting outputs. Here's how to picture it:

Think of inputs passing through different operations, which then produce outputs. For example, when the input set (a, b, c) goes through the operation F, the result is:

Now, when the same input set moves through the operation G, the output will be:

Linear algebra is a powerful tool for solving complex systems of linear equations. By using matrix operations, you can transform a data vector into an identity matrix. This process helps you find the values of the unknown variables at the output.

Simplifying Linear Algebra Operations

Linear algebra operations can seem tricky, but let's simplify them. Suppose you have three inputs. You can create 1-operation matrices to perform tasks more easily:

This will make the process much clearer and easier to follow.

The Adder

In simple terms, the adder is just the sum of a, b, and c. You are adding these values together to get the total.

The Average

The average, on the other hand, is calculated by adding up the numbers and then dividing by how many numbers there are. For example, if you have the numbers a, b, and c, you would find the average like this:

This gives you the middle value of the numbers, showing a central tendency.

Now try these easy one liners:

Now, let’s bring everything together in one matrix. This results in what’s called the “identity matrix,” which simply copies the three inputs directly to the three outputs.

Next, let’s see what happens when we reorder the inputs. For example, (x, y, z) could change to (x, z, y).

 This shows how the order of the inputs can be rearranged and how that affects the results.

The example above shows how an input can be doubled. In simpler terms, it works like this: when you use the identity matrix (2.1), and treat the inputs like vector coordinates, the operations matrix starts to behave like vectors.

Geometric Interpretations of Operations

In linear algebra, different operations can be visualized geometrically:

• Scale: This operation either shrinks or enlarges the inputs.

• Skew: It alters the inputs, making some of them bigger or smaller.

• Rotate: This operation creates new coordinates based on the original ones.

• Flip: It turns inputs into their negative counterparts.

These geometric actions on multiplication help reshape vector spaces. Keep in mind that vectors are simply representations of the data you need to adjust.

Practice and Perseverance

The key to mastering linear algebra is consistent practice. If you use the techniques we’ve discussed and practice daily, you’ll start feeling more comfortable with the subject.

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It’s not as hard as it seems. How you approach the subject can make all the difference. Look at it from a fresh perspective, and soon you’ll understand the concepts of linear algebra. 

We’re here to support you along the way.