ML Math - Scalars and Vectors

In Machine Learning, we often need to calculate the "length" of a vector, known as its Magnitude or Euclidean Norm.

Consider a 2D feature vector$v$ representing the position of a data point:

Using the Pythagorean theorem formula $\|\mathbf{v}\| = \sqrt{x^2 + y^2}$, calculate the magnitude of $\mathbf{v}$.

In a computer vision model, you have a vector$\mathbf{u}$ representing the RGB color values of a pixel. To dim the brightness of the pixel by half, you multiply the vector by a scalar $k = 0.5$

If you apply the scalar multiplication $k\mathbf{u}$, what is the resulting vector?

In a robotics simulation, a robot is being pushed by two different motors simultaneously.

• Motor A applies a force vector: $\mathbf{a} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$
• Motor B applies a force vector: $\mathbf{b} = \begin{bmatrix} -1 \\ 4 \end{bmatrix}$

What is the Resultant Vector ($a + b$) that represents the final direction and strength of the robot's movement?

The Dot Product is a fundamental operation in Neural Networks. It takes two vectors of the same length and returns a single scalar.

Calculate the dot product ( $\mathbf{a} \cdot \mathbf{b}$) for the following vectors:

Formula: $(a_1 \times b_1) + (a_2 \times b_2) + (a_3 \times b_3)$

If you multiply a non-zero vector $\mathbf{v}$by a negative scalar (for example, $-2$), the resulting vector will have a different magnitude, but it will point in the exact same direction as the original vector.

• True or
• False

You are creating a Machine Learning model to predict house prices. Each house is represented as a feature vector $x \in \mathbb{R}^3$, where the order of components is:

1. Square footage
2. Number of bedrooms
3. Age of the house (years)

If House A is 2000 sq ft, has 3 bedrooms, and is 10 years old, which representation correctly shows this data point as a vector?

In your Python environment, you have two vectors:

• Vector A is a 5-dimensional vector.
• Vector B is a 3-dimensional vector.

What will happen if you attempt to perform vector addition ($A + B$)

In Machine Learning, a Unit Vector is a specialized vector whose magnitude (Euclidean length) is exactly 1.

• True or
• False

Arrange the following steps in the correct order to calculate the Dot Product of two vectors:

In a K-Nearest Neighbors (KNN) algorithm, we measure the similarity between two items by calculating the distance between their vectors. If the distance is small, the items are similar.

Calculate the distance between Point A and Point B using the Euclidean distance formula:

Given:

Note: Use the calculator embedded in the quiz right panel when needed (Available on desktop webpage).

In Machine Learning, we update model weights ($\mathbf{w}$) by subtracting a "gradient" vector ($\mathbf{g}$) scaled by a learning rate ($\mathbf{\alpha}$).

If your current weight vector is $\mathbf{w}$  and your update vector is $\mathbf{u}$, what is the new weight vector?

Two vectors are considered Orthogonal (perpendicular, forming a 90-degree angle) if their Dot Product is exactly zero.

Consider these two vectors:

True or False: These vectors are orthogonal.

In standard Linear Algebra, you cannot mathematically add a Scalar (a single number) to a Vector. However, programming libraries like NumPy allow this through a concept called Broadcasting.

What happens when you execute the following Python code?

import numpy as np
vec = np.array([10, 20, 30])
result = vec + 5

In Machine Learning formulas, we often see the symbol $\mathbf{v}^T$. This represents the Transpose of vector $\mathbf{v}$.

If $\mathbf{v}$ is a Column Vector:

What does the transpose $\mathbf{v}^T$ look like?

A Linear Combination is the sum of vectors that have been scaled by different scalars.

Calculate the resulting vector for the following combination:

Note: Use the calculator embedded in the quiz right panel when needed (Available on desktop webpage).

The Dot Product $\mathbf{a} \cdot \mathbf{b}$ tells us about the alignment of two vectors.

If the Dot Product is positive, what does it suggest about the angle between the two vectors?

In Machine Learning code (using NumPy), there is a big difference between Element-wise Multiplication and a Dot Product.

Which of the following statements are correct? (Select all that apply)

In Clustering (like K-Means), we find the center of a group of points by calculating their average. To find the midpoint between two vectors, we add them and multiply by the scalar 0.5.

Find the midpoint of:

Note: Use the calculator embedded in the quiz right panel when needed (Available on desktop webpage).

In Natural Language Processing (NLP), we often use Sparse Vectors to represent documents.

True or False: A vector is considered "Sparse" if most of its components are zero.

• True or
• False

In Machine Learning, we optimize models by moving weight vectors in the opposite direction of the gradient. This is the core of "Learning."

Arrange the steps in the correct order to perform a single weight update:

In many ML algorithms (like SVMs), we need to Normalize our feature vectors to have a magnitude of exactly 1.

Arrange the steps to transform a raw vector into a Unit Vector:

An artificial neuron performs a specific sequence of arithmetic to produce an output.

Arrange the operations in the order they occur inside a single neuron:

In Natural Language Processing, a common way to find the Cosine Similarity between two word-embeddings (vectors) is to first "Normalize" them and then find their alignment. This prevents long documents from having an unfair advantage over short ones.

Arrange the steps in the strict logical order required to find the Cosine Similarity score using this normalization approach:

Note: Use the calculator embedded in the quiz right panel if you are practicing this with real numbers!

If you have a feature vector $\mathbf{v}$ and you multiply it by the scalar $0$, the result is always the Zero Vector (a vector where all components are 0).

• True or
• False

In the context of Machine Learning, the term Magnitude (or Norm) refers to a Vector quantity because it describes the distance from the origin.

• True or
• False

You are debugging a Python script using NumPy. Which of the following operations are mathematically valid between two vectors $\mathbf{u}$ and $\mathbf{v}$ that both have the shape (4, 1)? (Select all that apply)

Which of the following can be represented as a Vector in a Machine Learning dataset? (Select all that apply)

Perform the following calculation and determine the final Scalar result:

Note: Use the calculator embedded in the quiz right panel when needed (Available on desktop webpage).

In Data Science, we try to avoid "Redundant" features. If one vector is just a scalar multiple of another, they are Linearly Dependent.

Which of the following pairs are Linearly Dependent? (Select all that apply)