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Intersection and Union of Subspaces Intersection of Two Subspaces is a Subspace Proof: Let \( S \) and \( T \) be two subspaces of a vector space \( V \) over a field \( F \). We shall prove that \( S \cap T \) is also a subspace of \( V \). Let \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Then \( x, y \in S \) and \( x, y \in T \). Since \( S \) and \( T \) are subspaces of \( V \), we have: \[ \alpha x + \beta y \in S \] \[ \alpha x + \beta y \in T \] Therefore, \[ \alpha x + \beta y \in S \cap T \] Thus, \( \alpha x + \beta y \in S \cap T \) for all \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Hence, \( S \cap T \) is a subspace of \( V \). Union of Two Subspaces Need Not Be a Subspace Now, by giving an example we shall prove that the union of two subspaces need not be a subspace. Example: Let \( V = \mathbb{R}^3 = \{(x, y, z) : x, y, z \in \mathbb{R}\} \) be a vector space over the field \( \mathbb{R} \) of all rea...

Liouville's Theorem Art-15. Liouville's Theorem This theorem is very important, if you read this theorem carefully then some of the theorems coming up are based on this on which you can apply it and there are many questions where you can apply this theorem and you can solve those questions with Liouville's Theorem . Statement: If a function \( f(z) \) is analytic for all finite values of \( z \) and is bounded, then \( f(z) \) is constant. Proof: Let \( z_1, z_2 \) be any two points of the \( z \)-plane. Draw a circle \( C \) with center at the origin and radius \( R \), enclosing the points \( z_1 \) and \( z_2 \) so that: \( |z_1| < R \) and \( |z_2| < R \) Since \( f(z) \) is bounded, there exists \( M > 0 \) such that \( |f(z)| \leq M \) on \( C \). By Cauchy's integral formula, we have \( f(z_1) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_1} \, dz \) and \( f(...

It is very important to understand Rule of Divisibility because it is a way that a student can do faster calculations in their exam.  Divisibility rule are the shortcut way to determine that if a number is divisible by another number or not. It can enhance our calculation speed and accuracy and we can find the answer with in a second. Rule of Divisibility by 2: If the unit digit of a number is any 0,2,4,6 and 8 then that number will be completely divisible by 2.  Let's take a numbers:  98973232 8878438 893830 78733456 8768762494 Above, all the number are completely divisible by 2.  Let's take a number: 98973232 ÷ 2 = 49486616. Now, let's take a number where one's place have different number which is not divisible by 2.  For Example: 897987 ÷ 2 = 448993.5. Let's Understand from YouTube Rule of Divisibility by 3: If the sum of all the digits of the given number is completely divisible by...

  In this page, we are going to analysis the important topic or theorem in Complex Analysis.    Fundamental Theorem of Algebra in Complex Analysis is a very important topic to understand that can enhance your Higher Mathematics knowledge. Fundamental Theorem of Algebra in Complex Analysis Statement: If f(z) =  a 0 z n + a 1 z n-1 + ... + a n , a 0 ≠ 0 be a polynomial in z, then f(z) = 0 has at least one root in z-plane. Proof: If possible, let f(z) = 0 has no root in z-plane. Therefore, f(z)  ≠ 0 for any value of z in z-plane. We define, because, f(z) is analytic and f(z)  ≠ 0 therefore, F(z) is analytic for all values of z. Also, |F(z)| --> 0 as |z| -->  ∞  therefore, F(z) is bounded therefore, By Liouville's Theorem, F(z) is constant. ⇒ f(z) is constant, which is contradiction [because, f(z) is not constant polynomial] therefore, f(z) = 0 has at least one root in z-plane. Explanation in YouTube:

Exam Questions What is the unit digit in the number of (2137) 753 ? a) 1 b) 2 c) 7 d) 9 Answer: To find the unit dight in the number(2137) 753 , we need to focus on the circle of unit digits in the number are raised to powers. Here, we have to focus on the when the power rised and which number is in the unit place. The unit digit of 2137 is 7. When the number 7 will rise the power as following patterns are repeats. 7 1 has a unit digit of 7 7 2 has a unit digit of 9 7 3 has a unit digit of 3 7 4 has a unit digit of 1 7 5 has a unit digit of 7 SInce, the pattern repeats every four powers, now we can find the reminder when 753 is divided by 4: 753/4 = 188 and with give a reminder of 1. This means that the unit digit of (2137) 753 will be same as unit dight of (2137) 1 ∴ the unit digit of (2137) 753 . That is option c is correct.