University Examinations 2023/2024
LINEAR ALGEBRA I
QUESTION ONE (30 MARKS)
a) Show that (0, 1, 0) is a linear combination of (3, 0, 9), (0, 5, 1) and (-1, 0, 0) (3 Marks) b)
Given the points (1, -2, -3)
(i) Find the general vector on the plane through these points ( 3 Marks)
(ii) Find the equation of the plane containing the points (4 Marks)
c) Given that = (1, −3, 2) and = (−1, −2, −3) evaluate;
(i) . (2 Marks) (ii) x
(2 Marks) d) Let = (, , ); + +
≥ 0 in . Show that G is a subspace of (4 Marks)
e) Determine if vectors (1, 2, 3), (4, 5, 6) and (2, 1, 0) are linearly independent (4 Marks)
f) Let : ℝ be linear . Prove that ker f is a subspace of . (4 Marks)
g) Show that the vectors = (1, 2, 3), = (1, 0, 2) and = (1, 1, 0) form a basis for the
vector space . (4 Marks)
QUESTION TWO (20 MARKS)
a) Find the angle between (2, 2,1) and (1, 4, 8) (2 Marks)
b) Determine C so that that = (3, −4, 6) " = (3, −5, $) are perpendicular. (2 Marks)
c) Show that 2% − 3& + 4’ and −4% + 6& − 8’ are parallel. (2
Marks)
d) Prove that the diagonals of a rhombus are orthogonal to each other. (5 Marks)
e) (i) Find the equation of the plane which passes through (-1, 2, 3) and is perpendicular
to the planes 2 − 3 + 4 = 1 and 3 − 5 + 2 = 3 (4 Marks)
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LINEAR ALGEBRA I
QUESTION ONE (30 MARKS)
a) Show that (0, 1, 0) is a linear combination of (3, 0, 9), (0, 5, 1) and (-1, 0, 0) (3 Marks) b)
Given the points (1, -2, -3)
(i) Find the general vector on the plane through these points ( 3 Marks)
(ii) Find the equation of the plane containing the points (4 Marks)
c) Given that = (1, −3, 2) and = (−1, −2, −3) evaluate;
(i) . (2 Marks) (ii) x
(2 Marks) d) Let = (, , ); + +
≥ 0 in . Show that G is a subspace of (4 Marks)
e) Determine if vectors (1, 2, 3), (4, 5, 6) and (2, 1, 0) are linearly independent (4 Marks)
f) Let : ℝ be linear . Prove that ker f is a subspace of . (4 Marks)
g) Show that the vectors = (1, 2, 3), = (1, 0, 2) and = (1, 1, 0) form a basis for the
vector space . (4 Marks)
QUESTION TWO (20 MARKS)
a) Find the angle between (2, 2,1) and (1, 4, 8) (2 Marks)
b) Determine C so that that = (3, −4, 6) " = (3, −5, $) are perpendicular. (2 Marks)
c) Show that 2% − 3& + 4’ and −4% + 6& − 8’ are parallel. (2
Marks)
d) Prove that the diagonals of a rhombus are orthogonal to each other. (5 Marks)
e) (i) Find the equation of the plane which passes through (-1, 2, 3) and is perpendicular
to the planes 2 − 3 + 4 = 1 and 3 − 5 + 2 = 3 (4 Marks)
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