(Solved): (a) ........ Find the reduced echelon form and determine the rank of the matrix \[ A=\left(\begin{ ...

(a) ........ Find the reduced echelon form and determine the rank of the matrix \[ A=\left(\begin{array}{cccc} 2 & 0 & 2 & 0 \\ 1 & 1 & 1 & 1 \\ 5 & -4 & 5 & -4 \\ 0 & 1 & 0 & 1 \end{array}\right) . \] Use this to determine the dimension of the subspace of \( \mathbb{R}^{4} \) spanned by \( (2,0,2,0) \), \( (1,1,1,1),(5,-4,5,-4),(0,1,0,1) \). (b) ... Give an example of a \( 3 \times 5 \) matrix in reduced echelon form with no zero rows. (c) \( (1,4,7) \ldots \) Determine whether or not the vectors \( (1,2,-3),(-1,0,5) \) and i. form a spanning set for \( \mathbb{R}^{3} \); ii. are linearly independent; iii. form a basis for \( \mathbb{R}^{3} \). (d) ???. Write down a basis for \( \mathbb{R}^{n} \) for any positive integer \( n \) (a justification is not required). (e) L. Give an example of a subset of \( \mathbb{R}^{2} \) which is not a subspace of \( \mathbb{R}^{2} \) and contains infinitely many elements. Briefly justify your answer.