Cryptocurrency Q&A Can a matrix be injective?

Can a matrix be injective?

ShintoSpirit ShintoSpirit Sat Oct 19 2024 | 6 answers 1449
I'm wondering if a matrix can possess the property of being injective. I understand that functions can be injective, but I'm not sure if this concept applies to matrices as well. Can a matrix be injective?

6 answers

SakuraBloom SakuraBloom Mon Oct 21 2024
In the realm of mathematics, matrices are fundamental structures that encapsulate numerical data in a rectangular array. When discussing properties of matrices, a key concept is the row reduced form, denoted as Ared, of a given matrix A. This transformed matrix serves as a cornerstone in analyzing various attributes of A.

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mia_anderson_painter mia_anderson_painter Mon Oct 21 2024
One such attribute is injectivity, a term borrowed from set theory to describe functions that map distinct elements of the domain to distinct elements of the codomain. In the context of matrices, injectivity refers to the ability of A to preserve the distinctness of its column vectors.

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HanRiverWave HanRiverWave Mon Oct 21 2024
To determine if a matrix A is injective, we turn to its row reduced form Ared. This process simplifies A by applying a series of row operations, transforming it into a form that reveals its essential structural properties.

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Sofia Sofia Mon Oct 21 2024
If, upon inspection of Ared, we find that every column contains a leading 1 (the first non-zero element in each column), this indicates that A is injective. The presence of leading 1s in every column signifies that no column of A can be expressed as a linear combination of the others, thus preserving the distinctiveness of its column vectors.

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CryptoWizardry CryptoWizardry Sun Oct 20 2024
Conversely, if Ared has at least one column without a leading 1, it implies that the corresponding column in A can be linearly represented by the other columns. This lack of independence among the columns of A undermines its ability to be injective, as distinct input vectors may map to the same output vector.

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