Let $ f : X \rightarrow Y $ be a function. If B is subset of y then its inverse image $ f^-B $ is the subset of x define by $ f^-B = {x : f(x) \in B} $ Now prove the following. $ i. B_1 \subset B_2 \Rightarrow f^-B_1 \subset f^- B_1 $ […]

# Linear Algebra

## If x belongs to the domain then does f(x) belong to the range?

Let $ F : x \Rightarrow y $ be a function. Then prove that $i. A_1 \subset A_2 \Rightarrow F(A_1) \subset F(A_2)$ $ii. F(\cup_i A_i) = \cup_i F(A_i) $ $iii.F(\cap_i A_i) \subset \cap_i f(A_i)$ Solution $ Let \ f: x \rightarrow y $ be a function. i. $Let\ y \in f(A_1)$ $Then\ \exists x \in […]

## Is the empty set a subspace of every vector space? Justify

Question: a. Is the empty set a subspace of every vector space ? Justify and let s = ${(a, b, (a + b):a, b \in R)}$ . Is a subspace of $R^3$ under the usual operation. Question: b. Let I = (-a , a) a > 0 be an open interval in R let v […]

## Writing a Linear Regression Class from Scratch Using Python

Linear Regression Introduction Before there was any ML algorithms, there was a concept and that was regression. Linear Regression is considered as the process of finding the value or guessing a dependent variable using the number of independent variables. Take for a example:- predicting a price of house using variables like, size of house, age […]