Linear algebra is not only needed in many math courses but also a must have tool in many other areas like physics, engineering, computer science, statistics, natural and social sciences.

Compared with calculus, linear algebra can be abstract sometimes. We have resources for basic and as well as advanced linear algebra mainly for undergraduate students and beginning graduate students. Some questions are created by considering various types of mistakes made by students.

Linear algebra developed from the needs to solve systems of linear equations, which have geometrical meaning themselves, for example equations of lines, planes and higher dimensional planes. As an example, we solve two systems of linear equations:

\[\begin{cases} 3 x_{11}+2x_{21} &= \ 4\\ x_{11}+x_{21} &= \ 3\end{cases} \text{ and } \begin{cases} 3 x_{12}+2x_{22} & = \ 3\\ x_{12}+x_{22} &= \ 2 \end{cases} \]

We can just deal with one matrix and do it all at once. Row reduce the combined matrix to get

\[\left(\begin{array}{rr}3 & 2 \\ 1 & 1\end{array}\middle|\begin{array}{rr}4 & 3 \\ 3 & 2\end{array}\right) \longrightarrow \left(\begin{array}{rr}1 & 0 \\ 0 & 1\end{array}\middle|\begin{array}{rr}-2 & -1 \\ 5 & 3\end{array}\right).\] %\arrow{1}

From the above, we can write the solutions for the two systems all at once: \[\begin{cases} x_{11} &= \ -2\\ x_{21} &= \ 5\end{cases} \text{ and } \begin{cases} x_{12} & = \ -1\\ x_{22} &= \ 3 \end{cases}. \]

There are lots of online resources that can help you to "cheat" when doing your homeworks, but do calculations by hand still plays an important role for active learning. Here is an example of online matrix calculations:

https://matrixcalc.org/en/#Gaussian-elimination%28%7B%7B1,2,3%7D,%7B3,4,5%7D,%7B1,1,1%7D%7D%29

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