Finding the dimension of range and null space given a matrix and determining bases

Question

I am attempting to solve the following problem.

I am given the matrix A=\begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \\ \end{pmatrix}

Specifically I am asked to: (1) find the dimension of the range of this matrix, (2) give a range for this subspace, (3) find the dimension of the null space of this matrix, and (4) give a basis for this subspace.

I just keep staring at the problem but can't seem to put my finger on how to tackle it since I am not given a linear transformation.

Thanks in advance for the help!

Answer 1

Note that the $\ker A$ has dimension $3$ and is spanned by vector $(1,0,0,-1) ,(0,1,0,-1) , (0,0,1,-1)$ while $ Im A$ is spanned by vector $(1,1,1,1)$ Indeed to find $\ker A$ you have to solve a linear system $$\begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \\ \end{pmatrix}\cdot \begin{pmatrix} X_1 \\ X_2 \\ X_3 \\ X_4\\ \end{pmatrix}= vector zero$$ Therefore you find that $X_1+X_2+X_3+X_4=0$