What Level of Math is Linear Algebra?

What Level of Math is Linear Algebra?

Math & Technology

Linear Algebra is one of the most important divisions in Mathematics.

Many useful concepts, such as vector space, linear transformation and so on, have been popular in geometry and analysis for a long time. Among them, probably the most well-known concept is matrix, a rectangular array of numbers arranged in rows and columns. Matrices are helpful when we deal with linear algebra problems. Sometimes, solving problems with tricks of matrices are much easier than calculating directly: 

We can see that the base change matrix can be very powerful, as it can make the coordinate matrix much tidier by changing to a more straightforward basis of the vector space. Sometimes it’s hard to find such a matrix, and that would be the time for us to use this matrix trick.

However, we have to admit that this method doesn’t work for all case. The property of inverse of matrix is useful, but not all matrices have that property. They have to satisfy several requirements, such as being a square matrix and determinant non-zero. Therefore, if the matrix we want doesn’t have an inverse, this method doesn’t work. Also, we have to admit that finding the inverse of the matrix is also difficult. For example, the matrix like this is still tough work:

However, the method does provide us an insight of a different way to find the base change matrix, and sometimes it does make the problem easier. And by thinking more, it can help us outside the region of linear algebra.  This way of thinking, always trying to find an alternative method, will benefit us all the time. 

Written by Yujun Song

Math & Technology

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What Level of Math is Linear Algebra?