Core of matrix (pure translatio)

In summary, the core of a matrix refers to the set of all vectors that become zero when multiplied by the matrix. It is related to the rank of the matrix, where the dimension of the core is equal to the number of columns minus the rank. The core can be empty if the matrix has full rank, and it is calculated by performing Gaussian elimination and identifying the pivot columns. The core has important applications in linear algebra and other fields of mathematics, providing insights into the structure and properties of a matrix.
  • #1
maros522
15
0
Hello all,

I have problem with name of type of matrix. The definition is next:
The core of matrix A is collection of vectors x, for which is valid Ax=0.

Does anybody know the name of this type in english language. Example will be also good.

Thank You
 
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  • #3
It's usually called the kernel or the null space of A
kerA ={x | Ax=0}
 
  • #4
Thank You, it is exactly what I want.
 

FAQ: Core of matrix (pure translatio)

What is the core of a matrix?

The core of a matrix refers to the set of all vectors that become zero when multiplied by the matrix. In other words, it is the solution space of the homogeneous system of equations represented by the matrix.

How is the core of a matrix related to its rank?

The rank of a matrix is the dimension of the vector space spanned by its columns. The dimension of the core of a matrix is equal to the number of columns minus the rank. In other words, the core represents the linearly dependent combinations of the columns of the matrix.

Can the core of a matrix be empty?

Yes, the core of a matrix can be empty if the matrix has full rank. This means that there are no linearly dependent combinations of its columns, and therefore, no vectors that become zero when multiplied by the matrix.

How is the core of a matrix calculated?

The core of a matrix can be calculated by performing Gaussian elimination on the matrix and then identifying the pivot columns. The columns that are not pivot columns will form the basis for the core of the matrix.

What is the significance of the core of a matrix?

The core of a matrix has important applications in linear algebra and other fields of mathematics. It can be used to determine the rank of a matrix, solve systems of linear equations, and find solutions to optimization problems. It also provides insights into the structure and properties of a matrix.

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