What is the course about?
Spatial Languages is a series of courses in the Humanities department at City Lit. Each aims to unpack a mathematical theory in a way that is accessible and relevant to philosophers, creative artists and others. Mathematics offers great conceptual clarity and intuitive pictures that can be valuable when forming views about the visual.
At their simplest, vectors represent locations in space. A vector becomes most interesting when we consider it as a element of a system – a so-called “vector space” – that encapsulates the structure of locatedness in a very specific way. Vector spaces arose in this way in connection with physics and engineering in the nineteenth century, although the intuitive ideas had been around at least a century earlier.
The notion of a vector space turns out to be the crucial idea required for defining “dimension” in a robust and precise way. In a sense, dimensionality is one of the key themes of the whole subject, and we take it as our guiding subject.
Once the concrete idea of a vector space has been clarified, it is easy to see it as an abstract “fragment of structure” and seek other, non-spatial phenomena that share the same intrinsic properties. Thus we find vector spaces of thousands of dimensions arising quite naturally in the context of financial data, and even infinite-dimensional spaces.
Today the study of vector spaces – under the uninspiring title of “linear algebra” – is a cornerstone of science, technology and computing. An appreciation of it is essential to any mathematically-informed engagement with concepts of space and time.
This course proceeds by way of drawing, visualization and definition of concepts; we carry out only a few very easy calculations, and then only as a way to make our ideas explicit.
What will we cover?
• Concrete notions of vector as spatial location, in 2D and 3D.
• Dimension, spanning sets, basis and the notion of linear independence
• Strategies for visualizing higher-dimensional spaces
• Duality and the notion of covectors
• Linear transformations and their representations by matrices
• The algebra of vectors and vector spaces, including direct sum, tensor product and wedge product.
• Applications of vectors in a variety of fields
• Abstract systems of numbers (briefly).
What will I achieve?
By the end of this course you should be able to...
• Give precise definitions of the central objects of the study of vector spaces, and state some of its key results.
• Provide visual interpretations of these definitions and results and explore them through drawing.
• Perform simple symbolic manipulations using vectors and covectors.
• Apply the language of vectors to a wide variety of problems and identify when this language is at work in different contexts.
What level is the course and do I need any particular skills?
This course has no specific prerequisites – in particular, no mathematical skills or knowledge will be assumed.
How will I be taught, and will there be any work outside the class?
We will use a mixture of presentation and discussion in class. Some optional reading for between classes will sometimes be provided. We also encourage exploring the ideas on the course with the free software Geogebra, which is available for most modern computers or smartphones, but this is not a requirement.
Are there any other costs? Is there anything I need to bring?
No, all required materials will be supplied during the course.
When I've finished, what course can I do next?
You might be interested in HP101 Spatial languages: exploring connectedness, which starts in April 2020.