Akanksha Kapoor

Subspace Definition
A subset W of a vector space V is a subspace if it satisfies three key properties:
- Contains the Zero Vector: The zero vector of V must be in W.
- Closed under Addition: If u and v are in W, then u + v must also be in W.
- Closed under Scalar Multiplication: If u is in W and c is a scalar, then cu must also be in W.
Verif... moreication of the Given Set
The set defined as {(x1, x2, x3) ∈ R³ : x1 + 2x2 + 3x3 = 0} can be examined for these properties.
1. Contains the Zero Vector
- The zero vector (0, 0, 0) satisfies the equation:
0 + 2(0) + 3(0) = 0.
- Therefore, the zero vector is included in this set.
2. Closed under Addition
- Let (x1, x2, x3) and (y1, y2, y3) be two vectors in the set such that:
x1 + 2x2 + 3x3 = 0 and y1 + 2y2 + 3y3 = 0.
- Adding these vectors gives:
(x1 + y1) + 2(x2 + y2) + 3(x3 + y3) = (x1 + 2x2 + 3x3) + (y1 + 2y2 + 3y3) = 0 + 0 = 0.
- Hence, the set is closed under addition.
3. Closed under Scalar Multiplication
- For any vector (x1, x2, x3) in the set and scalar c:
c(x1, x2, x3) = (cx1, cx2, cx3).
- We check:
cx1 + 2(cx2) + 3(cx3) = c(x1 + 2x2 + 3x3) = c(0) = 0.
- Thus, the set is closed under scalar multiplication.
Conclusion
Since all three properties are satisfied, the given set is indeed a subspace of R³.

Isomorphism in Group Theory

In mathematics, specifically in group theory, isomorphism is a concept that relates two groups. An isomorphism between two groups means that the two groups have the same structure and can be considered essentially the same. In this case, we are given two groups g and h, and we need to determine if g/h is isomorphic to any other group.

The... more Given Groups

Let's start by understanding the given groups:

- Group g: g = Z4 × Z2
- Z4 is the cyclic group of integers modulo 4, denoted as {0, 1, 2, 3} with addition modulo 4.
- Z2 is the cyclic group of integers modulo 2, denoted as {0, 1} with addition modulo 2.
- Z4 × Z2 is the direct product of Z4 and Z2, which consists of ordered pairs (a, b) where a belongs to Z4 and b belongs to Z2. The group operation is defined component-wise, i.e., (a, b) + (c, d) = (a + c, b + d).

- Group h: h = (2, 0)
- h is a single element in the group, which represents an ordered pair (2, 0) in Z4 × Z2.

Isomorphism to Another Group

To determine if g/h is isomorphic to another group, we need to understand the quotient group g/h and its properties.

- Quotient Group g/h: g/h is the set of all left cosets of h in g. Each left coset is formed by taking an element g' from g and multiplying it on the left by the inverse of h. Mathematically, g/h = {g' * h | g' belongs to g}.

Since h = (2, 0), we can calculate the left cosets of h in g as follows:

- (0, 0) + h = {(0, 0) + (2, 0)} = {(2, 0)}
- (1, 0) + h = {(1, 0) + (2, 0)} = {(3, 0)}
- (2, 0) + h = {(2, 0) + (2, 0)} = {(0, 0)}
- (3, 0) + h = {(3, 0) + (2, 0)} = {(1, 0)}

Therefore, the quotient group g/h = {(2, 0), (3, 0), (0, 0), (1, 0)}.

Isomorphism Analysis

To determine if g/h is isomorphic to any other group, we need to consider the structure and properties of g/h.

- Size of g/h: The size of g/h is equal to the number of left cosets, which in this case is 4.

- Structure of g/h: The elements of g/h are (2, 0), (3, 0), (0, 0), (1, 0). We can observe that g/h is a cyclic group of order 4, as it contains all possible powers of (2, 0) in its