Mathematics for Data Science 2

01 Extra Activity 1

Exercise Questions 🔥 Exercise Solutions 🧯 Good morning! Here in India on this Wednesday, let’s break down this excellent problem on the PageRank algorithm. It’s a great example of how linear algebra is used to model and solve real-world problems like ranking webpages. This is a multi-step problem, so we’ll go through each part in order.

02 Extra Activity 2

Exercise Questions 🔥 Exercise Solutions 🔬 Here are the detailed solutions and concept explanations for each question from your images. ❓ Question 1 Consider the linear transformation T which scales each of the standard basis vectors by a non-zero constant $\lambda$. In the geogebra project, this corresponds to moving the vectors $\mathbf{Te1}$ and $\mathbf{Te2}$ so that the arrowheads are on $(\lambda, 0)$ and $(0, \lambda)$, respectively, for some $\lambda \in \mathbb{R} \setminus {0}$ of your choice. Which of the following statements are true for such a linear transformation? $\Box$ The matrix of T with respect to the standard bases for the domain and codomain is a scalar matrix. $\Box$ If $\lambda < 0$, the angle between $\mathbf{u}$ and $T(\mathbf{u})$ is $180^\circ$, for all $\mathbf{u} \in \mathbb{R}^2 \setminus {(0, 0)}$. $\Box$ If P is the square with vertices $(0, 0), (1, 0), (1, 1)$ and $(0, 1)$, then its image T P has area equal to $\lambda$ square units. $\Box$ If $\lambda > 0$, then there is a non-zero vector $\mathbf{u}$ such that $T(\mathbf{u}) = \mathbf{u}$. 💡 Answer and Concepts The correct statements are the first two:

03 Graded Assignment 10

Exercise Questions ❓ Solutions 🃏 Here are the detailed step-by-step solutions for the questions extracted from the uploaded images. Section 1: Bayesian Statistics (The “Fill in the Blanks” Problem) Problem Overview: We are analyzing a customer transaction success rate $p$. Likelihood: Bernoulli($p$) with $n=40$ trials and $k=28$ successes. Prior: Beta Distribution with Mean $\mu = 0.4$ and Variance $\sigma^2 = 0.02$. Goal: Find prior parameters, posterior distribution, and posterior mean. Step 1: Determine Prior Parameters (Blanks A, B, C) Concept: For a Beta distribution $\text{Beta}(\alpha, \beta)$:

04 Graded Assignment 2

Exercise Questions 🔥 Exercise Solutions 🧯 Good morning! Here in India on this Wednesday, let’s work through this comprehensive set of questions. They cover a wide range of important topics in linear algebra and its applications, from solving systems of equations to modeling real-world problems.

05 Graded Assignment 3

Exercise Questions 🧯 Exercise Solutions 🧪 Good morning! Here in India on this Wednesday, let’s explore this excellent collection of questions. They cover a wide range of important topics in linear algebra, from the fundamental axioms of vector spaces to real-world applications using matrices.

06 Graded Assignment 4

Exercise Questions ❓ Exercise Solutions 🧪 Good morning! Here in India on this Monday, this is a great set of questions covering many essential topics in linear algebra. Let’s work through them. Core Concepts: Matrices, Vector Spaces, and Linear Independence Rank of a Matrix: The rank is the maximum number of linearly independent rows (or columns) in the matrix. It represents the dimension of the row space or column space. Elementary row operations do not change the rank. Vector Space: A set of objects (vectors) that can be added together and multiplied by scalars, obeying certain axioms (like closure, associativity, identity elements, etc.). $\mathbb{R}^n$ and the set of $m \times n$ matrices ($M_{m \times n}(\mathbb{R})$) are common examples. Subspace: A subset of a vector space that is itself a vector space under the same operations. It must contain the zero vector and be closed under addition and scalar multiplication. Basis: A set of linearly independent vectors that span the entire vector space (or subspace). Dimension: The number of vectors in a basis for a vector space (or subspace). Linear Independence: A set of vectors is linearly independent if the only way to make a linear combination of them equal to the zero vector is by using all zero coefficients. Otherwise, they are linearly dependent. Span: The span of a set of vectors is the set of all possible linear combinations of those vectors. It forms a subspace. Question 1: Properties of Rank (from file image_743d9c.png) The Question: Which of the following statements are correct?

07 Graded Assignment 5

Exercise Questions ❓ Exercise Solutions 🔬 Hello! I can certainly help you with these linear algebra questions. Here is a detailed breakdown of each problem with the core concepts and step-by-step solutions. ❓ Question 1: Student Marks The marks obtained by Karthika, Romy and Farzana in Quiz 1, Quiz 2 and End sem (with the maximum marks for each exam being 100) are shown in Table M2W5G1.

08 Graded Assignment 6

Exercise Questions ❓ Exercise Solutions Here is a detailed breakdown of each question, including the core concepts and step-by-step solutions. ❓ Question 1 A function $T: V \to W$ between two vector spaces $V$ and $W$ is said to be a linear transformation if the following conditions hold: Condition 1: $T(v_1 + v_2) = T(v_1) + T(v_2)$ for all $v_1, v_2 \in V$. Condition 2: $T(cv) = cT(v)$ for all $v \in V$ and $c \in \mathbb{R}$.

09 Graded Assignment 7

Exercise Questions ❓ Exercise Solution 🟩 Here are the detailed answers and conceptual explanations for each of the questions you provided. ❓ Question 1: Identifying Inner Products Consider the vector spaces $V$ and functions $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ defined as follows: i) $V = \mathbb{R}^2$ and $\langle (x_1, x_2), (y_1, y_2) \rangle = x_1y_1 - x_2y_1 - x_1y_2 + 2x_2y_2$. ii) $V = M_{2 \times 2}(\mathbb{R})$ and $\langle A, B \rangle = Tr(AB)$. iii) $V = M_{2 \times 1}(\mathbb{R})$ and $\langle A, B \rangle = Tr(AB^t)$. iv) $V = \mathbb{R}^2$ and $\langle (x_1, x_2), (y_1, y_2) \rangle = x_1y_2 + x_2y_1$. Which of the following options is an inner product?

10 Graded Assignment 8

Exercise Questions ❓ Exercise Solutions 🟩 Here is a detailed solution and explanation for the matching problem. ❓ Problem Analysis You are asked to match sets of vectors (Column A) with their correct mathematical properties (Column B). The space is $\mathbb{R}^3$ (vectors with 3 components) and the inner product is the standard dot product: $\langle (x_1, x_2, x_3), (y_1, y_2, y_3) \rangle = x_1y_1 + x_2y_2 + x_3y_3$

11 Graded Assignment 9

Exercise Questions ❓ SOlutions Hello Aryan. These questions explore the visualization of multivariable functions, partial derivatives, and the fundamental definitions of function graphs and properties. Here are the step-by-step solutions for all 7 questions. Questions 1 - 4: Matching Functions to Graphs Concept: To match the functions, we analyze their shapes and symmetries:

12 Determinants (part 1)

https://youtu.be/A3fxp49I4U8

13 Determinants (part 2)

https://youtu.be/wejxX0YYYg4

14 Matrices

https://youtu.be/rnIDlZnrCc0 Of course! Here are the detailed solutions and explanations for each of the questions you provided.

15 Practice Assignment 1

16 Systems of Linear Equations

https://youtu.be/WzR4NKeLHMY Exercise Questions 🔥 Exercise Solutions 🧯 Of course! Here are the detailed solutions and explanations for the questions about systems of linear equations.

17 Vectors

https://youtu.be/1So2VV9Tm_A Exercise Questions 🔥 Exercise Solutions 🧯 Of course! Here are the detailed answers and concepts for each of the questions you provided.

18 Graded Assignment 1

Of course! Here are the detailed solutions and explanations for all the questions you provided. Question 1: System of Linear Equations Problem Suppose there are two types of oranges and two types of bananas available in the market. Suppose 1 kg of each type of orange costs ₹50 and 1 kg of each type of banana costs ₹40. Gargi bought x kg of the first type of each fruit, orange and banana, and y kg of the second type of each fruit, orange and banana. She paid ₹250 for oranges and ₹200 for bananas. Which of the following options are correct with respect to the given information?