[Math] Basis of Column Space of matrix that is in Row Reduced Echelon form

Question

I'm considerably new to proving things(and to Linear Algebra too).So, I hope someone would help me with this.

While proving Row Rank of Matrix = Column Rank of Matrix

Proof used the point that

For a Row Reduced Echelon Matrix, Basis of Column Space is just set of
columns that contain leading non-zero entries.

Can someone provide a proof of this (Hints would more be appreciated)?

Answer 1

For the proof you are looking for take a look to Prove that pivot columns of row reduced form of any matrix forms a basis in column space of that row reduced matrix. for the simpler case and here for the other Pivot columns of A are a basis for Col(A).

For the proof of this fundamental result and property of matrices you can take a look here Proofs that column rank = row rank.

For the intuition behind take a look here intuitive explanation why the row rank is equal to the column rank for a matrix.