MAT2611
ASSIGNMENT 3
LINEAR Algebra III
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 14
, Solutions:
Problem 1
We need to find a basis for the subspace of ℝ5 spanned by:
𝑣1 = (1,3,1,2, −1), 𝑣2 = (0,1,3,2, −1),
𝑣3 = (1,0,1,0, −1), 𝑣4 = (0,2,0, −1, −1).
Form a matrix with these vectors as rows and row-reduce to find the linearly
independent ones.
1 3 1 2 −1
0 1 3 2 −1
[ ]
1 0 1 0 −1
0 2 0 −1 −1
𝑅3 ← 𝑅3 − 𝑅1 :
1 3 1 2 −1
0 1 3 2 −1
[ ]
0 −3 0 −2 0
0 2 0 −1 −1
𝑅3 ← 𝑅3 + 3𝑅2 , 𝑅4 ← 𝑅4 − 2𝑅2:
Page 2 of 14
ASSIGNMENT 3
LINEAR Algebra III
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 14
, Solutions:
Problem 1
We need to find a basis for the subspace of ℝ5 spanned by:
𝑣1 = (1,3,1,2, −1), 𝑣2 = (0,1,3,2, −1),
𝑣3 = (1,0,1,0, −1), 𝑣4 = (0,2,0, −1, −1).
Form a matrix with these vectors as rows and row-reduce to find the linearly
independent ones.
1 3 1 2 −1
0 1 3 2 −1
[ ]
1 0 1 0 −1
0 2 0 −1 −1
𝑅3 ← 𝑅3 − 𝑅1 :
1 3 1 2 −1
0 1 3 2 −1
[ ]
0 −3 0 −2 0
0 2 0 −1 −1
𝑅3 ← 𝑅3 + 3𝑅2 , 𝑅4 ← 𝑅4 − 2𝑅2:
Page 2 of 14
