MAT2611 Assignment 9 2025 - Due August 2025

MAT2611
ASSIGNMENT 9 2025
DUE: 8 AUGUST 2025

, MAT2611
ASSIGNMENT 09
Due date: Thursday, 8 August 2025



1 = ( 2; 3) and v2
Problem 1. Consider the basis S = fv1 ; v2 g for R2 ; where v = ( 1; 0).
Let T : R2 ! R3 be the linear transformation such that
2

T (v1 ) = (2; 3; 1) and T (v ) = (1; 2; 4) :

Find a formula for T (x; y) ; and use that formula to …nd T (2; 3).




Problem 1
We are given a basis S = {v1 , v2 } for R2 , where:
​ ​




v1 = (−2, 3),
​ v2 = (−1, 0)
​




and a linear transformation T : R2 → R3 such that:

T (v1 ) = (2, 3, 1),
​ T (v2 ) = (1, 2, 4)
​




Step 1: Express (x, y) as a linear combination of v1 , v2 ​ ​




Let:

(x, y) = av1 + bv2 = a(−2, 3) + b(−1, 0) = (−2a − b, 3a)
​ ​




​




So we solve:

​
−2a − b = x and 3a = y
y
From the second equation: a = 3
​




Substitute into the first: ​




y 2y
−2 ( ) − b = x ⇒ b = −x −
3 3
​ ​




Step 2: Use linearity of T

T (x, y) = a ⋅ T (v1 ) + b ⋅ T (v2 )
​ ​




y 2y
= (2, 3, 1) + (−x − ) (1, 2, 4)
3 3
​ ​