Vectors study guide

1. Start with the type and index set

When interpreting a vector, identify both its entries and the field from which they are drawn. For example, [2.0, 3.14159, -1.0, 2.718281828] is a 4-vector over ℝ. Its function interpretation uses the set S = {0, 1, 2, 3} as the input set and ℝ as the co-domain. The index identifies which value is returned: 2 ↦ -1.0 means that input 2 maps to -1.0.

2. Translate between representations

The list, dictionary and function descriptions can encode the same vector. In Python, the list [2.0, 3.14159, -1.0, 2.718281828] corresponds to {0: 2.0, 1: 3.14159, 2: -1.0, 3: 2.718281828}. The dictionary makes the function interpretation explicit because each key is mapped to a value. A one-dimensional array is another representation: in VB.NET, Dim example(3) As Single describes a 4-entry array whose highest stated index is 3.

In an exam, explain the representation rather than simply naming it. For example, say that a list stores the entries in order, while a dictionary associates each index with its mapped value. Check that the number of entries agrees with the stated vector dimension.

3. Connect algebra and geometry

The vector [2.0, 3.0] can be viewed as an arrow from the origin to (2.0, 3.0). Vector addition achieves translation, so explain an addition question using movement or displacement as well as component values. Scalar-vector multiplication achieves scaling, so explain that the scalar changes the vector by scaling it. Keep these operations distinct: addition combines vector entries, whereas scalar-vector multiplication applies a scalar to a vector.

4. Recognise a convex combination

To decide whether an expression αu + βv is a convex combination of u and v, check both conditions: α and β must each be at least zero, and their sum must be one. For instance, coefficients 0.25 and 0.75 satisfy the required conditions. Coefficients that are negative, or coefficients whose sum is not one, do not satisfy the stated definition.

5. Calculate and explain a dot product

For u = [u1, …, un] and v = [v1, …, vn], multiply corresponding entries and add the results. With u = [2, 3] and v = [4, 5], the reasoning is (2 × 4) + (3 × 5) = 8 + 15 = 23, so u · v = 23. The operation is a scalar product: its result is a single value rather than another list. The specification identifies finding the angle between two vectors as an application of the dot product, so link a dot-product calculation to that application when interpreting an exam question.

Self-check

Can you state the field and dimension of a given vector? Can you write its function and dictionary interpretations? Can you describe the arrow for a two-dimensional vector? Can you distinguish translation from scaling? Can you test both conditions for a convex combination? Finally, can you multiply corresponding entries and add them without changing the order or omitting a term? If any answer is uncertain, rewrite the vector with indexed entries before continuing.