Prove dimension of row space and column space are same
Answers
Step-by-step explanation:
Row Space
The vector space generated by the rows of a matrix viewed as vectors. The row space of a n×m matrix A with real entries is a subspace generated by n elements of R^m, hence its dimension is at most equal to min(m,n). It is equal to the dimension of the column space of A (as will be shown below), and is called the rank of A.
The row vectors of A are the coefficients of the unknowns x_1,...,x_m in the linear equation system
Ax=0,
(1)
where
x=[x_1; |; x_m],
(2)
and 0 is the zero vector in R^n. Hence, the solutions span the orthogonal complement Oc to the row space Rs in R^m, and
dimOc+dimRs=m.
(3)
On the other hand, the space of solutions also coincides with the kernel (or null space) of the linear transformation T:R^m->R^n, defined by
T(x)=Ax
(4)
for all vectors x of R^m. And it also true that
dimKer(T)+dimI(T)=m,
(5)
where Ker(T) denotes the kernel and I(T) the image, since the nullity and the rank always add up to the dimension of the domain. It follows that the dimension of the row space is
dimRs=m-dimOc=m-dimKer(T)=dimI(T),
(6)
which is equal to the dimension of the column space.
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