How Do You Map Matrices to Complex Numbers in Linear Algebra?

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SUMMARY

The discussion focuses on mapping matrices to complex numbers in linear algebra, specifically addressing the matrix $A$ defined by the property $A^2 = \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$. This property indicates a correspondence with the complex number $i$. The set $K$ consists of matrices of the form $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$, where $a$ and $b$ are real numbers. To establish that $K$ is isomorphic to $\Bbb C$, one must demonstrate that the mapping from $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ to $a+ib$ is a bijective isomorphism that preserves both addition and multiplication.

PREREQUISITES
  • Understanding of matrix operations and properties
  • Familiarity with complex numbers and their representation
  • Knowledge of isomorphism in algebra
  • Basic linear algebra concepts, including vector spaces
NEXT STEPS
  • Study the properties of isomorphic structures in linear algebra
  • Learn about the representation of complex numbers as matrices
  • Explore the concept of matrix multiplication and addition
  • Investigate the implications of $A^2 = -I$ in linear transformations
USEFUL FOR

Students and educators in mathematics, particularly those focused on linear algebra, complex analysis, and abstract algebra. This discussion is beneficial for anyone looking to deepen their understanding of the relationship between matrices and complex numbers.

Eric Machisi
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Can someone please help me solve Q2 in the attachment
 

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The matrix $A$ has the interesting property that its square is minus the identity matrix: $A^2 = \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$. That suggests that it should correspond to the complex number $i$.

In fact, $K$ consists of all matrices of the form $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ where $a$ and $b$ are real numbers. To prove that $K$ is isomorphic to $\Bbb C$ you should show that the map taking $\begin{bmatrix}a&-b\\b&a\end{bmatrix}$ to $a+ib$ is an isomorphism. That is, it is a bijective map that preserves addition and multiplication.
 
It is sufficient to map $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ to 1 and map $\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ to i.
 

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