MAT2611 Assignment 5 Semester 2 2026 - Due 28 August 2026

MAT2611 Assignment 5 2026




ASSIGNMENT 05
Due date: Friday, 28 August 2026


Problem 1. Suppose S = fv1 ; v2 g is a basis of R2 ; where v1 = ( 1; 2) and v2 = (2; 3).
Let T : R2 ! R3 be the linear operator for which

T (v1 ) = (1; 2; 1) and T (v2 ) = (0; 1; 3) :

Find a formula for T (x; y) ; and use that formula to …nd T (2; 1).




Problem 1

Given:
S = {v₁, v₂} is a basis of ℝ², where v₁ = (-1, 2) and v₂ = (2, -3).

T: ℝ² → ℝ³ is a linear operator for which:
T(v₁) = (1, -2, 1) and T(v₂) = (0, -1, 3)

Find a formula for T(x, y), and use that formula to find T(2, -1).


Solution:

Express an arbitrary vector (x, y) ∈ ℝ² as a linear combination of v₁ and v₂.

(x, y) = a·v₁ + b·v₂ = a(-1, 2) + b(2, -3)

This gives the system:
-a + 2b = x ... (1)
2a - 3b = y ... (2)

Solve for a and b.

From (1): a = 2b - x

Substituting into (2):
2(2b - x) - 3b = y
4b - 2x - 3b = y
b - 2x = y
b = 2x + y

, Use linearity of T.

T(x, y) = (3x + 2y)T(v₁) + (2x + y)T(v₂)



Then:
a = 2(2x + y) - x = 4x + 2y - x = 3x + 2y

Therefore:
(x, y) = (3x + 2y)v₁ + (2x + y)v₂

( Friedberg, Insel & Spence, Section 2.1, Theorem 2.6 - a linear transformation is completely
determined by its action on a basis)