This is a problem on linear dependent columns of the data matrix ${\bf X}$ used in multiple regression.
Assume the row dimension $n$ of ${\bf X}$ is greater than the column dimension $k$ of ${\bf X}$. ${\bf X}$ has full column rank of the $k$ columns of ${\bf X}$ are linearly independent.
Let ${\tt X}_1, \ldots,{\tt X}_k$ denote the $k$ columns of ${\bf X}$. The columns of ${\bf X}$ are linearly independent if $a_1{\tt X}_1+\cdots+a_k{\tt X}_k={\bf 0}_n$ implies that $a_1=\cdots=a_k=0$. The columns of ${\bf X}$ are linearly dependent if $a_1{\tt X}_1+\cdots+a_k{\tt X}_k={\bf 0}_n$ for at least one vector ${\bf a}=(a_1,\cdots,a_k)^T$ that is not the zero vector.
Let ${\tt A}_1, \ldots,{\tt A}_k$ denote the $k$ columns of the $k\times k$ matrix ${\bf A}={\bf X}^T{\bf X}$. The columns of ${\bf A}$ are linearly independent if $c_1{\tt A}_1+\cdots+c_k{\tt A}_k={\bf 0}_k$ implies that $c_1=\cdots=c_k=0$. The columns of ${\bf A}$ are linearly dependent if $c_1{\tt A}_1+\cdots+c_k{\tt A}_k={\bf 0}_k$ for at least one vector ${\bf c}=(c_1,\cdots,c_k)^T$ that is not the zero vector.
Please check your linear algebra text for the definition of column rank.
Notation: ${\bf 0}_s$ denotes the zero vector in $s$-dimensional space.


Part a)
Suppose the columns of ${\bf X}$ are linearly dependent based on the coefficient vector ${\bf a}=(a_1,\cdots,a_k)^T$, that is, $a_1{\tt X}_1+\cdots+a_k{\tt X}_k={\bf 0}_n$. Which of the following are correct statements and implications.
Possibly more than one item is correct.

A. ${\bf a}^T{\bf X}={\bf 0}_n$
B. ${\bf a}{\bf X}={\bf 0}_n$
C. ${\bf X}{\bf a}={\bf 0}_n$
D. the dimension of the null space of ${\bf X}^T{\bf X}$ is less than or equal to that of ${\bf X}$
E. ${\bf X}^T{\bf X}{\bf a}={\bf 0}_k$
F. the dimension of the null space of ${\bf X}^T{\bf X}$ is greater than or equal to that of ${\bf X}$
G. the column rank of ${\bf X}^T{\bf X}$ is less than or equal to the column rank of ${\bf X}$


Part b)
Suppose the columns of ${\bf A}={\bf X}^T{\bf X}$ are linearly dependent based on the coefficient vector ${\bf c}=(c_1,\cdots,c_k)^T$, that is, $c_1{\tt A}_1+\cdots+c_k{\tt A}_k={\bf 0}_k$. Which of the following are correct statements and implications.
Possibly more than one item is correct.

A. ${\bf c}{\bf A}={\bf 0}_k$
B. ${\bf X}{\bf c}={\bf 0}_k$
C. $({\bf X}^T{\bf X}){\bf c}={\bf 0}_k$
D. $({\bf X}{\bf c})^T({\bf X}{\bf c})=0$
E. ${\bf A}{\bf c}={\bf 0}_k$
F. ${\bf c}^T{\bf X}^T{\bf X}{\bf c}={\bf 0}_k$
G. ${\bf c}^T{\bf X}^T{\bf X}{\bf c}=0$
H. ${\bf A}{\bf c}={\bf 0}_n$
I. ${\bf X}{\bf c}={\bf 0}_n$
J. the column rank of ${\bf X}$ is less than or equal to the column rank of ${\bf X}^T{\bf X}$


Part c)
What is the column rank of

Part d)
What is the column rank of

Part e)
What is the row rank of

Part f)
Which of the following are ways to determine the column rank of ${\bf X}$?
There might be more than one correct answer.

A. The number of non-zero diagonal entries of ${\bf R}$ in the QR decomposition of ${\bf X}={\bf Q}{\bf R}$ where ${\bf Q}$ is an $n\times k$ matrix with orthogonal columns, and ${\bf R}$ is a $k\times k$ upper triangular matrix: qrobj=${\tt qr}({\bf X})$, Q=${\tt qr.Q}$(qrobj), R=${\tt qr.R}$(qrobj) in R
B. The number of non-zero singular values of ${\bf X}$: ${\tt svd}({\bf X})$ in R
C. Algorithm of (i) subtracting multiples of a column to get zeroes in first row and k-1 columns, (ii) then subtracting multiples of another column to get zeroes in the second row and k-2 columns, (iii) iterate to row k: column rank is the number of columns that are not the zero vector
D. max { s: column s is linearly independent of columns 1,...,s-1}
E. The number of positive eigenvalues of ${\bf X}^T{\bf X}$: ${\tt eigen}({\bf X}^T{\bf X})$ in R
F. None of the above



From the above, write at least two proofs for the statement: "the rank of ${\bf X}$ and ${\bf X}^T{\bf X}$ are the same".

Hint: