UNISA
ASSIGNMENT 09 FOR 2023 SOLUTIONS–
MAT2611
MAT 2611
Reading Time: 15 Minutes
Writing Time: TWO Hours
Permitted Materials: Non-programmable Calculators; Mathematical tables
Page 1 of 9 – MAT2611 (MAT 2611)
, ASSIGNMENT 09: SOLUTIONS
Due date: Friday, 11 August 2023
Problem 31. Suppose that S = {v1 , v2 } is a basis of R2 wherev1 = (1, −1) and v2 =
(2, 0). Let T : R2 −→ R2 be the linear operator for which
T (v1 ) = (3, 2) and T (v2 ) = (−1, 0)
Find a formula for T (x, y), and compute T (3, −2).
SOLUTION TO PROBLEM 31. Let v = (x, y) ∈ R2 be arbitrary then v can be
expressed as a linear combination of v1 and v2 where v1 = (1, −1) and v2 = (2, 0) and
T (v1 ) = (3, 2) and T (v2 ) = (−1, 0) so that
v = av1 + bv2
(x, y) = a(1, −1) + b(2, 0)
(x, y) = (a + 2b, −a)
Hence it follows that,
x = a + 2b and y = −a
2b = −a + x and a = −y
−a x
b= +
2 2
−(−y) x
b= +
2 2
y x
= +
2 2
T (x, y)
= T (v)
= T (av1 + bv2 )
= aT (v1 ) + bT (v2 )
= a(3, 2) + b(−1, 2)
y x
= −y(3, 2) + ( + )(−1, 0)
2 2
y x
= (−3y − − , −2y)
2 2
7y x
=(- − , −2y)
2 2
Is the required formula for T (x, y)
Moreover, we proceed to determine the formula for T (3, −2) as shown below
−7 3
T (3, −2) = ( (−2) − , −2(−2))
2 2
3
T (3, −2) = (7 − , 4)
2
11
T (3, −2) = ( , 4)
2
Page 2 of 9 – MAT2611 (MAT 2611)
ASSIGNMENT 09 FOR 2023 SOLUTIONS–
MAT2611
MAT 2611
Reading Time: 15 Minutes
Writing Time: TWO Hours
Permitted Materials: Non-programmable Calculators; Mathematical tables
Page 1 of 9 – MAT2611 (MAT 2611)
, ASSIGNMENT 09: SOLUTIONS
Due date: Friday, 11 August 2023
Problem 31. Suppose that S = {v1 , v2 } is a basis of R2 wherev1 = (1, −1) and v2 =
(2, 0). Let T : R2 −→ R2 be the linear operator for which
T (v1 ) = (3, 2) and T (v2 ) = (−1, 0)
Find a formula for T (x, y), and compute T (3, −2).
SOLUTION TO PROBLEM 31. Let v = (x, y) ∈ R2 be arbitrary then v can be
expressed as a linear combination of v1 and v2 where v1 = (1, −1) and v2 = (2, 0) and
T (v1 ) = (3, 2) and T (v2 ) = (−1, 0) so that
v = av1 + bv2
(x, y) = a(1, −1) + b(2, 0)
(x, y) = (a + 2b, −a)
Hence it follows that,
x = a + 2b and y = −a
2b = −a + x and a = −y
−a x
b= +
2 2
−(−y) x
b= +
2 2
y x
= +
2 2
T (x, y)
= T (v)
= T (av1 + bv2 )
= aT (v1 ) + bT (v2 )
= a(3, 2) + b(−1, 2)
y x
= −y(3, 2) + ( + )(−1, 0)
2 2
y x
= (−3y − − , −2y)
2 2
7y x
=(- − , −2y)
2 2
Is the required formula for T (x, y)
Moreover, we proceed to determine the formula for T (3, −2) as shown below
−7 3
T (3, −2) = ( (−2) − , −2(−2))
2 2
3
T (3, −2) = (7 − , 4)
2
11
T (3, −2) = ( , 4)
2
Page 2 of 9 – MAT2611 (MAT 2611)
