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In this linear algebra course, we'll look at what linear algebra is and how it relates to vectors and matrices. Then we discuss what vectors and matrices are and how to work with them, including the tricky problem of eigenvalues and eigenvectors, and how to use them to solve problems. Finally, we'll look at how to use them to do fun things with data sets, like how to rotate images of faces and how to extract eigenvectors to see how the Pagerank algorithm works. Since we point to data-d
In this linear algebra course, we'll look at what linear algebra is and how it relates to vectors and matrices.
Then we discuss what vectors and matrices are and how to work with them, including the tricky problem of eigenvalues and eigenvectors, and how to use them to solve problems.
Finally, we'll look at how to use them to do fun things with data sets, like how to rotate images of faces and how to extract eigenvectors to see how the Pagerank algorithm works.
Since our goal is data-driven applications, we'll be implementing some of these ideas in code, not just pencil and paper.
Towards the end of the course, you'll write code blocks and find Jupyter notebooks in Python, but don't worry, these will be fairly short, focus on concepts, and guide you if you haven't coded before.
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In this first module, we discuss how linear algebra is relevant to machine learning and data science. Then we will end the module with an initial introduction to vectors. At all times, we focus on developing your mathematical intuition, not on working on algebra or doing long examples with pencil and paper. For many of these operations, there are callable functions in Python that can do the addition; the point is to appreciate what they do and how they work so that when things go wrong or there are special cases, you can understand why and what to do.
In this module, we look at the operations we can do with vectors: finding the modulus (size), the angle between the vectors (point or inner product), and the projections of one vector onto another. We can then examine how the inputs describing a vector will depend on what vectors we use to define the axes: the base. That will then allow us to determine whether a proposed set of basis vectors is what is called 'linearly independent'. This will complete our examination of vectors, allowing us to move on to matrices in Module 3 and then begin solving linear algebra problems.
Now that we've seen vectors, we can move on to matrices. First, we will see how to use matrices as tools for solving linear algebra problems and as objects that transform vectors. We then look at how to solve systems of linear equations using matrices, which then leads us to look at inverse matrices and determinants, and to think about what the determinant really is, intuitively speaking. Finally, we will see cases of special matrices that mean that the determinant is zero or where the matrix is not invertible, cases in which algorithms that need to invert a matrix will fail.
In Module 3, we continue our discussion of arrays; We first thought about how to encode matrix multiplication and matrix operations using the Einstein Addition Convention, which is a notation widely used in more advanced linear algebra courses. Next, we look at how matrices can transform a description of a vector from one basis (set of axes) to another. This will allow us, for example, to discover how to apply a reflection to an image and manipulate images. We will also see how to construct a set of basis vectors convenient for performing such transformations. We will then write code to do these transformations and apply this work computationally.
The eigenvectors are particular vectors that are not rotated by a transformation matrix, and the eigenvalues are the amount by which the eigenvectors are stretched. These special 'proper things' are very useful in linear algebra and will allow us to examine Google's famous PageRank algorithm for rendering web search results. We will then apply this in code, which will wrap up the course.
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Imperial College London is one of the top ten universities in the world with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and marketing, harnessing science and innovation to address global challenges.
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