Math Problem Statement

kerjakan no 1

Kerjakan no 1

Solution

To solve question number 1, we are given a set of matrices M2×2(R)M_{2 \times 2}(\mathbb{R}) where each matrix is in the form:

(abcd),a,b,c,dR\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad a, b, c, d \in \mathbb{R}

The problem statement suggests that the set of all 2×22 \times 2 matrices with real entries forms a vector space under matrix addition and scalar multiplication. Let's analyze this.

Solution:

To verify if the set M2×2(R)M_{2 \times 2}(\mathbb{R}) forms a vector space, we need to check if it satisfies the following axioms of a vector space over the field R\mathbb{R}:

  1. Closure under Addition: If we add two matrices of this form, the result will still be a 2×22 \times 2 matrix with real entries. (abcd)+(efgh)=(a+eb+fc+gd+h)\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix} Since a+e,b+f,c+g,d+hRa+e, b+f, c+g, d+h \in \mathbb{R}, the result is a matrix in M2×2(R)M_{2 \times 2}(\mathbb{R}).

  2. Closure under Scalar Multiplication: Multiplying any 2×22 \times 2 matrix by a scalar kRk \in \mathbb{R} will yield another 2×22 \times 2 matrix with real entries. k(abcd)=(kakbkckd)k \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix} Since ka,kb,kc,kdRka, kb, kc, kd \in \mathbb{R}, the result is a matrix in M2×2(R)M_{2 \times 2}(\mathbb{R}).

  3. Zero Element: The zero matrix, (0000)\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, is in M2×2(R)M_{2 \times 2}(\mathbb{R}) and acts as the additive identity.

  4. Additive Inverse: For each matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}, there exists an additive inverse (abcd)\begin{pmatrix} -a & -b \\ -c & -d \end{pmatrix} such that their sum is the zero matrix.

Since these properties are satisfied, we can conclude that the set of all 2×22 \times 2 matrices with real entries, M2×2(R)M_{2 \times 2}(\mathbb{R}), indeed forms a vector space under matrix addition and scalar multiplication.

Would you like further details or have any questions?

Follow-Up Questions:

  1. What other types of matrices could form a vector space?
  2. How does matrix addition and scalar multiplication work in higher dimensions?
  3. Can we find a basis for M2×2(R)M_{2 \times 2}(\mathbb{R}) and determine its dimension?
  4. What is the role of the zero matrix in vector spaces?
  5. How can we verify vector space properties in more abstract sets?

Tip:

Always verify the basic properties (closure, identity, inverse, and distributive properties) to confirm if a set with defined operations forms a vector space.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Matrix Operations

Formulas

Matrix addition
Scalar multiplication for matrices

Theorems

Vector space axioms
Closure under addition
Closure under scalar multiplication

Suitable Grade Level

Undergraduate - Linear Algebra