define kernal in algebra
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A vector v is in the kernel of a matrix A if and only if Av=0. Thus, the kernel is the span of all these vectors.
Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0.
For example the kernel of this matrix (call it A)
[100201]
is the following matrix, where s can be any number:
⎡⎣0−s2s⎤⎦
Verification using matrix multiplaction: the first entry is 0∗1−s∗0+2s∗0=0 and the second entry is 0∗0−s∗2+2s∗1=0.
[100201]∗⎡⎣0−s2s⎤⎦=[00]
A related concept is that of image of a matrix A.
The dimensions of the image and the kernel of A are related in the Rank Nullity Theorem
Similarly, a vector v is in the kernel of a linear transformation T if and only if T(v)=0.
For example the kernel of this matrix (call it A)
[100201]
is the following matrix, where s can be any number:
⎡⎣0−s2s⎤⎦
Verification using matrix multiplaction: the first entry is 0∗1−s∗0+2s∗0=0 and the second entry is 0∗0−s∗2+2s∗1=0.
[100201]∗⎡⎣0−s2s⎤⎦=[00]
A related concept is that of image of a matrix A.
The dimensions of the image and the kernel of A are related in the Rank Nullity Theorem
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The beauty of a sentence that is translated into the language of algebra is that it can then be transformed using clearly defined properties and identities, such as the Order of Operations or the Identity, Associative, and Commutative Properties of Addition and Multiplication.
These transformed equations can answer questions and reveal new insights into the original statement.
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