Vectors and Their Notation

Overview

Recall questions on what a vector is, list and set notation for vectors, and the function interpretation.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is a vector?

   Model answer

   An ordered collection of values, all drawn from the same field, which can be thought of in three equivalent ways: as a list of numbers, as a function that maps each position to a value, and as a way of representing a geometric point or a displacement in space.

2. 2

   Write the list notation for a 4-vector over the reals with entries 5.0, 1.5, -2.0 and 0.25.

   Model answer

   [5.0, 1.5, -2.0, 0.25]. The entries are written in order between square brackets, separated by commas, and the order matters because each position of the list is a distinct component of the vector.

3. 3

   How is the set of all 4-vectors over the real numbers written?

   Model answer

   ℝ⁴ — the symbol for the real numbers with a raised 4. In general, the set of n-vectors over a field F is written Fⁿ, so ℝ³ is the set of 3-vectors over the reals.

4. 4

   Explain the function interpretation of a vector.

   Model answer

   A vector is seen as a function f: S → ℝ, where the domain S is the set of index positions and the co-domain is ℝ, the set of real numbers. The function maps each index to the entry stored at that index. For the 4-vector [5.0, 1.5, -2.0, 0.25] the domain is S = {0, 1, 2, 3} and the mapping is 0 ↦ 5.0, 1 ↦ 1.5, 2 ↦ −2.0, 3 ↦ 0.25. The symbol ↦ means 'maps to'.

5. 5

   In the function interpretation of a 4-vector over the reals, what are the domain and the co-domain?

   Model answer

   The domain is the set of index positions, S = {0, 1, 2, 3}. The co-domain is ℝ, the set of real numbers, because that is the field the entries are drawn from.

6. 6

   Why must all the entries of a vector be drawn from the same field?

   Model answer

   Because the operations defined on vectors — adding two vectors and multiplying a vector by a scalar — are carried out entry by entry using the arithmetic of that field, and the results must themselves be valid entries. If the entries came from different fields those operations would not be consistently defined. For a vector over ℝ, every entry is a real number.

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