# Mastering Linear Algebra - Key concepts like matrices, determinants and linear transformations

In this course, we'll provide a comprehensive overview of linear algebra, including key concepts like vector spaces, linear transformations, and matrix operations.

## What is included in a linear algebra course?

The following 28 topics are typically included in a linear algebra course

## What is linear algebra?

Linear algebra is a branch of mathematics that, among other things, studies equations of the form: $$a_1x_1 + a_2x_2 + \ldots + a_nx_n = b$$ You usually study linear algebra first on university, but there are those who get an easier introduction to the subject already during High School. Many people associate a course in linear algebra with vectors, matrices, linear systems of equations, lines and planes. All associations are correct and can also be deduced from the equation form above.

## FAQ

What is a line in linear algebra?If the equation is two-dimensional, it is a line: $$a_1x_1 + a_2x_2 = b$$

What is a plane in linear algebra?If the equation is three-dimensional, it is a plane: $$a_1x_1 + a_2x_2 + a_3x_3 = b$$

What is a system of equations?If we have several equations in the same form, these can be considered a system of equations. The definition of a solution to the system must solve all equations in the system. One system of equations can be rewritten to: $$A\vec{x} = \vec{b}$$ using the matrix ($A$) and the two vectors $\vec{x}, \vec{b}$.

What is a vector?A vector in linear algebra is usually treated as a group of numbers and is one coordinate representation. Several examples from different dimensions, i.e. number of coordinates, is: $$(1,2), (1,-4,2), (99, 104, 3, -7)$$ While a vector in physics is usually drawn as a arrow and represents a force with a direction and magnitude. The notion that a vector is a force usually follows in mathematics, but here vectors are studied in an abstract meaning. In practice, a vector can be very much more than a force in physics. A vector can be a list of information in programming, an image in image analysis, a stock price change over time in finance, and much more.

What is a cross product?The cross product is a calculation between two vectors in three dimensions, and the result is a third vector that is unique and orthogonal to the first two. The length of the resulting vector is equal to the area of the parallelogram that the two vectors span.

## What is linear algebra used for? - 7 practical use cases

Linear algebra is the cornerstone of everything we see in our everyday life. It's thanks to linear algebra we make Boeing fly, Tesla drive and Spotify play. Linear algebra is also the basis of machine learning which has a number of applications, such as that Siri recognizes your face, Alexa recognizes your voice and that H&M maximizes its sales online. But how can linear algebra be critical for all vastly different applications?

Students usually say that the course feels abstract, which to a certain extent correct. The advantage of an abstract tool is that only the imagination limits area of use.

### 1. Encryption

A smart way to protect the private information we send to each other is through encryption and decryption processes involving inverse matrices. In order to ensure that an eavesdropper cannot read a message sent electronically; we encrypt it, distorting and rearranging the symbols to make them look like nonsense. In order for the right person to read it, the message must then be decrypted at the reception.

Using a technique called the Hill Cipher, the sender and receiver agree on a matrix to use to encrypt messages. Since the recipient knows that, they can use the inverse of the matrix to decrypt the message

### 2. Computed tomography (CT scan)

A CT scan is a medical imaging system that shoots X-rays through a body from many different angles. Based on these X-ray images, we can construct a picture of what the body looks like on the inside. But have you ever wondered how the image is actually constructed?

The answer is of course with mathematics, especially with the so-called Radon transform. Radon transform is a type of integral transformation, which is a general linear transformation.

### 3. Digital photo filters

A pixel refers to a small region on your screen represented by three numbers between 0 and 255 that indicate the intensity of the red, green, and blue components, respectively. We use pixels to create digital images, and to change the appearance of an image, we adjust the values of its pixels.

Setting the colors to similar values creates a grayscale image, and increasing or decreasing them will result in a lighter or darker appearance, respectively. By changing the intensity of the three components of the constituent pixels, there are endless other ways in which images can be manipulated to enhance certain features.Matrix arithmetics enables us to do this efficiently. Consequently, it's part of the math to thank for the filters that make your Instagram posts look amazing.

### 4. Pricing within e.g. insurance

Red cars are overrepresented in traffic accident statistics, but why aren't the insurance costs for red cars higher than for other colored cars of the same model? When you dig a little deeper, we find that it is not the color red itself that is a risk factor on the roads, but the color is linked to other characteristics that are. Red is a common color for sports cars, which tend to have powerful engines and male drivers. They also tend to have a high price tag.

The cost of the insurance is linearly dependent on the value of the car and the risk of collision, i.e. if these go up, the insurance premium increases by a proportional amount. The probability that a car is red in color and its insurance cost depend on the same parameters, but do not directly affect each other.

### 5. Search engines

Computers often perform calculations on matrices, and it turns out that special types of matrices where many of their elements are zero make calculations both faster and more accurate. Datorer utfĂ¶r ofta

Larry Page and Sergey Brin, the founders of Google, knew everything about how computer arithmetic works and how to optimize it. This enabled the revolution in the search engine market that Google brought about, increasing both the frequency of relevant hits on the web per diversity compared to their competitors.

### 6. Face recognition

Face detection systems are used to highlight the differences between people's faces so that only the right person is given access. Such programs rely heavily on the concept of eigenvectors.

It turns out that human faces are linearly dependent on combinations of certain distinguishing features, such as hair color, nose size, distance between eyes, and so on. To properly construct these features, the system needs many images of people to learn from, but after it has described the important aspects of faces, a much smaller amount of special images can be used to reconstruct and compare any of the people.

These special images are called eigenfaces. The name comes from the fact that they are essentially eigenvectors of a matrix containing information about the facial features found among the set of given images. Corresponding eigenvalues give a measure of how important each eigenface is to distinguish between different people. Dessa speciella bilder kallas fĂ¶r egenansikten.

### 7. Self-driving cars

Just like human drivers, self-driving cars must constantly scan the roads for obstacles and road signs to navigate our streets safely. To be able to do that, the car is equipped with cameras that take snapshots of the surroundings at very short intervals. But how does the car know if the Volvo in front is driving nonchalantly on the road, or has stopped suddenly as a result of an accident?

The answer is linear transformations. An image of a car far away has a completely different pixel representation compared to a close-up of the same car. However, there is a linear relationship between the images, as the car itself does not change its appearance. Through linear transformations that zoom and rotate the image sequence, the self-driving algorithm can determine the behavior of the car in front and act accordingly.

## Is linear algebra hard?

Linear algebra is usually considered a difficult threshold for students to cross. In addition to the fact that it is a new world in mathematics that is presented, there is also a new language with a lot of inconsistent usage. In addition, English is usually an additional threshold for all students whose mother tongue is not English. The most difficult parts of linear algebra are usually linear transformations, change of basis and linear transformation and bases.

## Why is it called linear algebra?

Linear algebra is called "linear" because it deals with linear equations and linear transformations . A linear equation is an equation in which the highest power of the variable is 1. For example, $2x + 3 = 0$ is a linear equation, whereas $x^2 + 4x + 3 = 0$ is not. Linear equations can be represented graphically as straight lines.

A linear transformation is a transformation of a vector space that preserves the operations of vector addition and scalar multiplication. In other words, it is a function that maps one vector to another in such a way that the properties of vectors and scalars are preserved.

Linear algebra also deals with concepts such as matrices, determinants, eigenvalues, and eigenvectors, which are used to represent and manipulate linear equations and transformations. All of these concepts have special properties when it comes to linearity that is why the name is Linear Algebra.

## Good outline for linear algebra and short to-do list

We work hard to provide you with short, concise and educational knowledge. Contrary to what many books do.

## Get exam problems for old linear algebra exams divided into chapters

The trick is to both learn the theory and practice on exam problems. We have categorized them to make it extra easy.