Vectors Spaces can be studied without Matrices and Matrices can be studied without Vector Space.
In reality when we expand the study of both, we will find that Matrices can take 3 vector spaces.
The 3 vector spaces are row space, column space and null space. These spaces also have their basis.
Vector Spaces of Matrices
If A is an m×n matrix
The subspace of R^n spanned by the row vectors of A is called the row space of A.
The subspace of R^m spanned by the column vectors is called the column space of A.
The solution space of the homogeneous system of equation Ax = 0, which is a subspace of R^n, is called the nullspace of A.
Basis of Matrices Vector spaces
If a matrix R is in row echelon form,
The row vectors with the leading 1’s (i.e., the nonzero row vectors) form a basis for the row space of R.
The column vectors with the leading 1’s of the row vectors form a basis for the column space of R.
I hope if there is a unique solution for matrix AX=B, then we this unique solution form the basis for null space in R. Still doubtful, but not required to be investigated further. [Found it true, Finding a basis of the null space of a matrix , was present in tool and it maps to solution space as basis]
Rank of a matrix in terms of Vector Spaces
If Row and column space have equal dimensions dim(RS(A)) = dim(CS(A)) then rank(A) = dim(RS(A)) = dim(CS(A))
rank(A^T) = dim(RS(A^T)) = dim(CS(A)) = rank(A) Therefore rank(A^T ) = rank(A)
Nullity of Matrix
If A is an m×n matrix of rank r, then the dimension of the solution space of Ax = 0 is n – r.
nullity(A) = dim(NS(A))
nullity(A) =n - rank(A)= n-r
n=rank(A)+nullity(A) --> Rank Nullity Theorem.
Rank for solution finding
Rank we can obtain from row reduction by getting rref.
If A is an m×n matrix and rank(A) = r, then
Fundamental Space => Dimension
RS(A)=CS(AT) => r
CS(A)=RS(A^T) => r
NS(A) => n-r
NS(AT) => m-r
We can only use guassian elimination method to find rank and the get the system of equation without 0 values and replace dependent rows with substituted values, we can find the nullity. Then we can check whether rank + nullity provides dimension n of matrix (or total no. of rows).
In reality when we expand the study of both, we will find that Matrices can take 3 vector spaces.
The 3 vector spaces are row space, column space and null space. These spaces also have their basis.
Vector Spaces of Matrices
If A is an m×n matrix
The subspace of R^n spanned by the row vectors of A is called the row space of A.
The subspace of R^m spanned by the column vectors is called the column space of A.
The solution space of the homogeneous system of equation Ax = 0, which is a subspace of R^n, is called the nullspace of A.
Basis of Matrices Vector spaces
If a matrix R is in row echelon form,
The row vectors with the leading 1’s (i.e., the nonzero row vectors) form a basis for the row space of R.
The column vectors with the leading 1’s of the row vectors form a basis for the column space of R.
I hope if there is a unique solution for matrix AX=B, then we this unique solution form the basis for null space in R. Still doubtful, but not required to be investigated further. [Found it true, Finding a basis of the null space of a matrix , was present in tool and it maps to solution space as basis]
Rank of a matrix in terms of Vector Spaces
If Row and column space have equal dimensions dim(RS(A)) = dim(CS(A)) then rank(A) = dim(RS(A)) = dim(CS(A))
rank(A^T) = dim(RS(A^T)) = dim(CS(A)) = rank(A) Therefore rank(A^T ) = rank(A)
Nullity of Matrix
If A is an m×n matrix of rank r, then the dimension of the solution space of Ax = 0 is n – r.
nullity(A) = dim(NS(A))
nullity(A) =n - rank(A)= n-r
n=rank(A)+nullity(A) --> Rank Nullity Theorem.
Rank for solution finding
Rank we can obtain from row reduction by getting rref.
If A is an m×n matrix and rank(A) = r, then
Fundamental Space => Dimension
RS(A)=CS(AT) => r
CS(A)=RS(A^T) => r
NS(A) => n-r
NS(AT) => m-r
We can only use guassian elimination method to find rank and the get the system of equation without 0 values and replace dependent rows with substituted values, we can find the nullity. Then we can check whether rank + nullity provides dimension n of matrix (or total no. of rows).
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