Will the dimension of the image of a linear transformation always equal the dimension of the basis of the image of a linear transformation?
Will the dimension of the kernel of a linear transformation always equal the dimension of the basis of the kernel of the aforementioned linear transformation?
The following will be an implication of the rank-nullity theorem: Let $A$ be the matrix that defines the linear transformation.
"dim" means dimension
"im" means image of
Let $m$ = the number of columns of a matrix that defines a linear
Let dim(A) = the number of non linearly dependent columns or
nonredundant columns of A.
Let dim(ker $A$) be $m$-dim(im$A$).
So far, I have always assumed that
dim(ker $A$)=dim(basis of kernel of $A$) and dim(im $A$)=dim(image of basis of $A$).
Is my assumption correct?