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Most everyday mathematics consists of manipulating numbers one at a time. Formally, these single numerical values are called scalars. For example, in the expression for converting temperature, $$c = \frac{5}{9}(f - 32)$$, the values $$5$$, $$9$$, and $$32$$ are constant scalars, whereas the variables $$c$$ and $$f$$ represent unknown scalars.

Scalars

A programmer is adjusting the volume of a digital audio signal. The audio signal is represented as a list of thousands of numerical values. To decrease the volume, every value in the list is multiplied by a single number, `0.75`. The code for this operation looks like this: `new_audio_signal = 0.75 * original_audio_signal`. 

In this scenario, the `original_audio_signal` is a collection of many numbers. Analyze the role of the number `0.75` in this operation. What is the specific mathematical term for a single numerical quantity that is used to scale a collection of other numbers in this manner?

Role of Quantities in a Scaling Operation

Match each mathematical object with its defining characteristic. The goal is to distinguish between different types of quantities based on their structure.

Examine the following mathematical operation: `[y₁, y₂] = 5 * [x₁, x₂] + 2`. Based on this expression, identify the components that are represented as single numerical values which modify the other quantities.

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