“Why do I struggle to do my math homework on linear algebra?” “How can I solve my linear algebra homework faster?”
Do these thoughts run through your mind often while doing your linear algebra homework? Well, know that the most significant reason behind such thoughts is poor foundational knowledge. Linear algebra is a fundamental aspect of mathematics and central to many applications.
Therefore, if you want to score well in your homework and build solid knowledge for the long run, you will need to build a strong base in linear algebra.
And, while you can drop a “Please help me “do my math homework” request at reputed math homework help services, it is best to complement professional academic assistance with your hard work for holistic development.
Start developing a solid base in linear algebra with this handy & intuitive learning guide.
A Brief Introduction To Linear Algebra
Linear algebra is a sub-branch of arithmetic. It is a continuous form of mathematics with numerous applications through science & engineering.
This is because linear algebra offers a convenient, effective, and well-organized to represent any data mathematically. Linear algebra is central to computational mathematics, functional analysis, data science, machine, and deep learning.
Matrices and vectors are the two significant ways of representing data and manipulating it in linear algebra. Specific examples can elucidate things further.
As evident above, the inner product of rows with x operation and the combination of columns produce the same results.
- The inner product of multiplying one row at a time is known as the dot product or scalar product.
- The vector or vector product approach combines the two columns of the two matrices linearly. This is the most rudimentary operation in linear algebra!
From the example above, we can see that the linear combinations of matrices and vectors involve multiplying the columns in a matrix with scalars. The results are then added to obtain the answer.
Thus, for a matrix or vector A and a scalar x, the Ax is the linear combination of the columns of A with x.
Consequentially, this leads us to the idea of the column space of a matrix. In mathematics & statistics, the idea of any sample space denotes the range of values of a random variable.
Thus, the column space of a matrix or vector is all the different combinations of the column. In the context of the example above, any real numbers x1 and x2 are valid; that is, the column space of A also includes the results of Ax for all vectors x. The column space of any linearly combined matrix or vector can thus be infinitely many output vectors.
The above concepts are the basic foundations of linear algebra. Let us dive deeper.
Linear Algebra, Vector Spaces & Analytical Geometry
Vectors and vector spaces are essential in linear algebra. Therefore, visualization is a key to effective data representation and easier understanding, and analytical geometry & the rectangular coordinate system are used to depict vectors & vector spaces.
For a typical vector v,
Vn are the components of the vector. The number of variables required to represent the components of a vector is known as its dimensionality; two- or three-dimensional vectors can be represented in 2D and 3D rectangular coordinate spaces.
In the example in the preceding section, the Ax is thus a vector in the 3-dimensional real space. This real 3D space is R3, where R denotes real numbers and 3 denotes the dimensionality of the real space.
For the linear combinations of the matrix A & the vectors x below, all possible linear combinations of the columns form the column space, which is a part of the greater three-dimensional space.
So, if we plot all linear combinations, then the resultant vectors form a plane in 3D real space. The place comprises all the lines in the direction of (2,2,3), all the lines in the direction of (3,4,7), and the sum of any vector on one line and any other vector on another line.
Linear algebra is vast and can become quite complex & tedious at times. However, the above concepts form the base of all operations and advanced nuances in linear algebra. Master them and start practicing more from these great books on linear algebra.
However, if you think you could do it with professional assistance, place a “do my math homework” request with a reputed service to get help from capable “do my math homework” helpers”.
Best Self-Study Books In Linear Algebra
Introduction To Linear Algebra By Gilbert Strang
One of the best books on the subject, this is a must-read if you want to strengthen your ideas. It offers a comprehensive, erudite, and discursive discussion on linear algebra. In addition, it offers an expansive view of the entire subject and shines light upon major intersections with other math branches & prominent applications.
Professor Strang is a mathematician and professor at the Massachusetts Institute of Technology. He takes a rigorous approach to understanding linear algebra, using endless examples to elucidate everything.
As a result, there is enough material in the book for an entire one-year course on linear algebra. In addition, MIT offers loads of supplementary material, including Professor Strang’s video lectures, interactive demonstrations, examinations, and problem sets. There is also a solution manual.
Linear Algebra & Its Applications By David C. Lay, Steven R. Lay, and Judi J. McDonald
Another great book to sharpen your linear algebra concepts, it takes a no-frills and no-nonsense approach to the subject. The book is relatively long and offers much more than an introductory course on linear algebra.
The explanations in the book use highly simplified models to help readers understand concepts. This helps readers gain a quick and easy understanding of the subject.
The book also thoroughly looks at linear algebra’s applications in many fields, such as science, engineering, computer application, businesses, etc. While the concepts & applications are not discussed rigorously, the book provides a good understanding of how linear algebra works and why it is so important.
Linear Algebra by Friedberg, Insel, & Spence
The standard choice for undergraduate students across the United States of America, this book is quite challenging and takes a much more applied approach to the subject.
It is excellent for self-studying thanks to the numerous intuitive & enlightening problems within. Many problems may seem a bit tough but require a little bit of intelligent problem-solving that can unravel the complexities.
The problems build upon each other, with easier problems presenting concepts necessary to solve more challenging problems.
Those are three excellent books to start practicing linear algebra and ace all your math homework on the subject. And, with that, we wrap up this write-up.
Build your basics to score better in all your linear algebra math homework. If you think you need some professional assistance, then do drop a “do my math homework Online” request at a reputed math homework help service.
Author-Bio: John Millar is a mathematics professor at a leading public research university in the USA. He is also a part-time tutor at Assignmenthelp.us, a leading academic service provider, and addresses hundreds of “math homework help” student requests every month.