Unit 1 Vectors, matrices, and linear maps
Vectors can represent displacements in space, but in this course they will more often represent data or model parameters. A word, document, image, or user can be encoded as a vector. Matrices then act on vectors, and the matrix-vector product gives the first large family of functions we study: linear maps.
Big questions. How do vectors represent data, how do matrices act on vectors, and what is special about linear functions?
Learning outcomes. By the end of this unit, students should be able to:
Core outcomes.
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U1-LO1. Compute and interpret vector norms, dot products, distances, orthogonality, and cosine similarity.
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U1-LO2. Represent objects or data as vectors and explain what cosine similarity and Euclidean distance measure differently.
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U1-LO3. Perform basic matrix operations and check dimensions for matrix, vector, and matrix-product expressions.
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U1-LO4. Compute \(Ax\) and interpret it both as row dot products and as a linear combination of the columns of \(A\text{.}\)
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U1-LO5. Identify, construct, and interpret matrix representations of linear transformations.
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U1-LO6. Compute small matrix products and explain how multiplication represents composition of linear maps.
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U1-LO7. Distinguish linear maps from affine maps such as \(\mathbf{x}\mapsto W\mathbf{x}+\mathbf{b}\text{.}\)
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U1-LO8. Interpret simple code, tables, or diagrams involving cosine similarity, attention-style weighted averages, and applying one matrix to many inputs.
Toolbox skills. Compute vector norms, distances, dot products, cosine similarities, matrix sums, scalar multiples, transposes, matrix-vector products, small matrix products, and simple weighted averages.
Application and computation outcomes. Interpret Lab U1
[U1-LO1, U1-LO2, U1-LO3, U1-LO4, U1-LO5, U1-LO6, U1-LO7, U1-LO8 | C+R+T | Core] by reading computations involving vector similarity, document retrieval, attention-style weighted averages, matrix-vector products, applying one matrix to many points, composition, affine maps, and short code outputs.
