What is Linear Algebra?

Imagine you’re a builder, tasked with constructing a sturdy bridge across a river. You need to plot straight lines and calculate angles to ensure the bridge stands tall and strong.

This is where linear algebra steps in! It offers you a toolbox filled with matrices and vectors to map out your design in a mathematical space.

Linear algebra is a type of math that solves straight-line problems. It uses grids of numbers called matrices to show these problems in vector space. Basically, it’s about how straight lines and vectors work in math.

Linear algebra is really important in fields like engineering and physics. It helps us figure out various natural phenomena.

The main things we look at in linear algebra are straight-line problems, matrices, and spaces. Understanding these ideas is key to getting the basics of this subject. In this article, we will cover:

• Linear-Algebra topics
• Resources to learn Linear Algebra
• 3 types of Linear Algebra
• Difference between Linear-Algebra and Algebra
• Importance of learning Linear Algebra
• Importance of learning Linear Algebra

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Linear Algebra topics

Linear algebra covers three main areas: linear equations, matrices, and vectors. The areas are interconnected and understanding them is crucial for mastering linear algebra. Let’s break down the topics within each category:

Linear Equations

A linear equation is a foundational concept of linear algebra. It’s a mathematical expression written in a standard form, like a1x1 + a2x2 + … + anxn. In Linear equations, you work with variables. 

Here are the key topics covered under linear equations:

• Linear Equations in One Variable
• Linear Equations in Two Variables
• Simultaneous Linear Equations
• Solving Linear Equations
• Solutions of a Linear Equation
• Graphing Linear Equations
• Applications of Linear Equations – Straight lines.

Vectors

Vectors in linear algebra involve various operations like multiplication and addition. You can calculate quantities like velocity using vectors. Important topics under vectors include:

• Types of Vectors
• Dot Product
• Cross Product
• Addition of Vectors

Matrices

Matrices are rectangular arrays used to organize data. They’re represented as Am×n, where m is the number of rows and n is the number of columns. Matrices are useful in expressing linear equations concisely. Here are the main topics covered under matrices:

• Matrix Operations
• Determinant
• Transpose of a Matrix
• Types of Matrices

What are the 3 types of Linear Algebra?

The three types of Linear Algebra are as follows:

Elementary Linear Algebra

Elementary Linear Algebra introduces you to the fundamentals of Linear Algebra. It includes vectors and matrix operations but at a simpler level.

Right below are the important terms from Elementary Linear Algebra:

• Scalars – Scalars are quantities that have magnitude but no direction. They’re typically real numbers and are used to define vector spaces.
• Vectors – Vectors are elements in a vector space with both direction and magnitude.
• Vector Space – A vector space consists of vectors.
• Matrix – A matrix is a rectangular array. It encloses organized information in rows and columns. Many properties of linear algebra can be expressed using matrices.
• Matrix Operations – These include basic arithmetic operations like addition, subtraction, and multiplication. The operations are performed on matrices.

To strengthen your algebra basics before diving into linear algebra, explore the best apps to learn algebra for effective practice.

Advanced Linear Algebra

After mastering the basics, students move on to advanced concepts in linear algebra. It focuses on linear equations, vectors, and matrices but at an advanced level.

Here are the important terms used in advanced linear algebra:

• Linear Transformations – These functions transform vectors from one vector space to another. While transforming the vector space they preserve the initial linear structure.
• Inverse of a Matrix – When multiplied by the original matrix, the inverse of a matrix results in the identity matrix.
• Eigenvector – An eigenvector is a non-zero vector. It changes only by a scalar factor (eigenvalue) when a linear transformation is applied to it.
• Linear Map – A linear map is a type of mapping that maintains vector addition and scalar multiplication properties.

Applied Linear Algebra

Applied linear algebra is typically introduced at a graduate level. It is taught in fields such as applied mathematics, engineering, and physics. 

It focuses on integrating elementary and advanced linear algebra concepts with practical applications. Here are key topics of Applied Linear Algebra used in fields like engineering, physics, and computer science: 

• Vector Norms – Measures the length or magnitude of a vector in a vector space. It provides a way to quantify the size of vectors.
• QR Factorization – Decomposes a matrix into the product of an orthogonal matrix (Q) and an upper triangular matrix (R). It is useful for solving systems of linear equations and eigenvalue problems.
• Schur’s Complement – A method used to simplify large matrices by partitioning them into smaller blocks. It is particularly useful in numerical analysis and optimization algorithms.

These topics are just a few examples of the many applications of applied Linear algebra. There can be more topics depending on the field it is being used for.

💡Read What is Algebra? to learn about Algebra curriculum and topics in detail.

Difference between Linear Algebra and Algebra

Though Linear Algebra is a branch of Algebra, they are not the same. The main difference between Linear Algebra and Algebra is here:

Linear Algebra	Algebra
Linear Algebra is only a “subset” of Algebra that deals with linear equations.	Algebra has many “subsets” of which Linear Algebra is just a part.

Resources to learn Linear Algebra

Learning Linear Algebra requires you to study algebraic concepts in depth. Reliable learning resources help you understand the topics better. You can use these resources for:

1. Revising a Linear Algebra lesson
2. Assessing your Linear Algebra skills
3. Learning a specific Linear Algebra topic in depth

You must choose resources that match your learning style and academic goals. Here are the reliable resources to help you learn Linear Algebra:

Free Linear Algebra worksheets

Practicing worksheet questions will help you assess your algebra skills. It will also improve your efficiency in solving problems. The questions in these worksheets are based on important formulas and concepts from your lessons. 

Here are popular websites to find free Pre-Algebra worksheets:

1. Kuta Software 

2. Math Worksheets 4 Kids 

3. Kidsmart Education   

Free online lessons for Pre-Algebra 

Online videos, audio lectures, various articles, and practice problems help you grasp your lessons anytime, anywhere.

Two reliable platforms where you can find free online lessons for Linear Algebra are

1. KhanAcademy: For course-specific lessons.

2. YouTube: For exploring the course in detail with math experts.

The popular YouTube channels that have videos on Pre-Algebra lessons are

• MIT OpenCourseWare

• Kyle Broder

• Geek’s Lesson

Linear Algebra books 

If you are looking for detailed books on Linear Algebra, here are the top suggestions:

1. Linear Algebra Done Right

2. Practical Linear Algebra for Data Science

3. Linear Algebra (Dover Book on Mathematics)

Linear Algebra tutoring

At times you need personalized and professional support from a tutor. If you need personalized help with your Pre-algebra concepts, homework, and test prep, you can consider these platforms:

1. Wiingy: For affordable and quality 1-on-1 tutoring, homework help and test-prep with Linear Algebra tutors.

2. Chegg: For homework help.

3. The Princeton Review: For test prep.

💡Wondering if Algebra is a hard math course? Read – Is Algebra Hard?

Importance of Linear Algebra in education and career

While some students take Linear Algebra because it is a part of their math curriculum, some opt for it because it interests them. Different people can take Linear Algebra for different reasons. Learning Linear Algebra is important because it –

Lays foundation for learning advanced math

According to the Perspectives and Reflections on Teaching Linear Algebra, Linear Algebra should not just be a higher mathematics course. 

Introducing students to Linear Algebra in early grades will lay a strong foundation for advanced math courses like Calculus. Without knowing linear algebra well, it’s hard to understand these other math topics.

Is used in STEM fields

The Mathematical Teaching Research Journal says Linear algebra is really important for engineering students to learn. It helps solve lots of problems in engineering, physics, and even other areas of math. 

For example, engineers use it to design bridges and circuits, while physicists use it to study how things move. Learning linear algebra opens up opportunities to work in different fields and solve real-world problems.

Helps in data analysis

With computers becoming the most important tool in corporates, knowing linear algebra is handy. It helps us analyze data, build machine learning models, and make computer programs run faster. It also helps you learn coding languages like Python.   

🥇Looking for the best tips to learn Algebra? Read: How to Learn Algebra?

Key takeaways on “What is Linear Algebra?”

Linear algebra is math that deals with lines, shapes, and spaces using numbers and grids. It’s essential in fields like engineering and physics to understand natural phenomena. Linear Algebra covers linear equations, matrices, and vectors. It can be divided into Elementary, Advanced, and Applied courses.

Resources like worksheets, online lessons, and books help in learning Linear Algebra smoothly. You can also get help from Linear Algebra tutors for professional support in concept clearing, homework help, and test prep.

Mastering linear algebra sets the foundation for advanced math and is valuable in STEM fields like engineering and data analysis.

📖Recommended Reading:

• In What Grade Should You Take Algebra 1 and Algebra 2?
• What is Algebra 1?
• What is Algebra 2?
• What is Pre-Algebra?

Looking to Learn Linear Algebra? Book a Free Trial Lesson and match with top Linear Algebra Tutors for Concepts, Homework Help, and Test Prep.

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