Relationship between # dimensions in image and kernel of linear transformation called A and # dimensions in basis of image and basis of kernel of A

Question

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
  transformation.

• 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?

Answer 1

Loosely speaking, the dimension of the image equals the number of elements in the basis of that image. Likewise for the kernel. Hence, the answer to your question, "Is my assumption correct?" is "no" if you want to use terminology correctly.