Math Problem Statement

selesaikan soal diatas

Perhatikan soal berikut! Himpunan matriks 2x2 dengan unsur-unsur bilangan riil dan operasi penjumlahan matriks serta perkalian skalar matriks merupakan ruang vektor atas lapangan bilangan riil. Selesaikan soal diatas.

Solution

The question asks about the set of all 2×22 \times 2 matrices with real entries, denoted as M2×2(R)M_{2 \times 2}(\mathbb{R}), and states that under matrix addition and scalar multiplication, this set forms a vector space over the field of real numbers.

To analyze this, we need to verify that M2×2(R)M_{2 \times 2}(\mathbb{R}) satisfies the properties of a vector space over R\mathbb{R}.

Solution:

  1. Vector Space Properties: A set forms a vector space over a field if it satisfies the following conditions with respect to addition and scalar multiplication:

    • Closure under addition: For any two matrices A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B=(efgh)B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}, their sum A+B=(a+eb+fc+gd+h)A + B = \begin{pmatrix} a + e & b + f \\ c + g & d + h \end{pmatrix} is also a 2×22 \times 2 matrix with real entries, so the set is closed under addition.

    • Associative property of addition: Matrix addition is associative, meaning for any matrices AA, BB, and CC, we have (A+B)+C=A+(B+C)(A + B) + C = A + (B + C).

    • Existence of an additive identity: The zero matrix O=(0000)O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} serves as the additive identity, as A+O=AA + O = A for any matrix AA.

    • Existence of additive inverses: For every matrix A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, there exists an additive inverse A=(abcd)-A = \begin{pmatrix} -a & -b \\ -c & -d \end{pmatrix} such that A+(A)=OA + (-A) = O.

    • Closure under scalar multiplication: For any scalar αR\alpha \in \mathbb{R} and matrix A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the product αA=(αaαbαcαd)\alpha A = \begin{pmatrix} \alpha a & \alpha b \\ \alpha c & \alpha d \end{pmatrix} is also a 2×22 \times 2 matrix with real entries.

    • Distributive properties: Scalar multiplication is distributive over matrix addition and scalar addition:

      • α(A+B)=αA+αB\alpha (A + B) = \alpha A + \alpha B
      • (α+β)A=αA+βA(\alpha + \beta) A = \alpha A + \beta A
    • Compatibility of scalar multiplication with field multiplication: For any scalars α,βR\alpha, \beta \in \mathbb{R} and any matrix AA, α(βA)=(αβ)A\alpha (\beta A) = (\alpha \beta) A.

    • Existence of a multiplicative identity: The scalar 11 acts as a multiplicative identity, as 1A=A1 \cdot A = A for any matrix AA.

  2. Conclusion: Since M2×2(R)M_{2 \times 2}(\mathbb{R}) satisfies all these properties, it is indeed a vector space over the real numbers R\mathbb{R}.

Would you like more detailed explanations for any of these properties?


Related Questions:

  1. What is the dimension of M2×2(R)M_{2 \times 2}(\mathbb{R}) as a vector space over R\mathbb{R}?
  2. How can we determine a basis for M2×2(R)M_{2 \times 2}(\mathbb{R})?
  3. Can M2×2(R)M_{2 \times 2}(\mathbb{R}) be extended to form a different kind of algebraic structure, such as a ring or algebra?
  4. What is the role of the identity matrix in M2×2(R)M_{2 \times 2}(\mathbb{R}) with respect to matrix multiplication?
  5. How does the vector space M2×2(R)M_{2 \times 2}(\mathbb{R}) relate to the concept of linear transformations on R2\mathbb{R}^2?

Tip:

Remember that each element of M2×2(R)M_{2 \times 2}(\mathbb{R}) can be represented as a linear combination of four basis matrices, which simplifies many vector space computations.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Vector Spaces
Linear Algebra

Formulas

Matrix addition formula: A + B = \begin{pmatrix} a + e & b + f \\ c + g & d + h \end{pmatrix}
Scalar multiplication of a matrix: \alpha A = \begin{pmatrix} \alpha a & \alpha b \\ \alpha c & \alpha d \end{pmatrix}

Theorems

Vector Space Axioms

Suitable Grade Level

Undergraduate