MAT2611 ASSIGNMENT 06 Due 27 June 2025

MAT2611
ASSIGNMENT 06
Due 27 June 2025

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MAT2611
ASSIGNMENT 06
Due date: Friday, 27 June 2025



Problem 1. Let T1 (x 1; x2; x3) = (x1 + x 2; x1 x 2; x3) and T2 (x 1; x2; x3) = (x1; x2 + x 3; x3) :

(a) Find the standard matrix for1 Tand T2:

(b) Find the standard matrix for1 T T 2 and T2 T 1:

(c) Use the matrices obtained in part (b) to …nd formulas
1 (T 2 (xT1; x2; x3)) and T2 (T 1 (x 1; x2; x3)).
for

[10 marks]


Problem 2. Find the standard matrix for the matrix operator de…ned by the equation and determine whether
the operator is one-to-one.
w 1 = x 1 + 2x2 x 3
w 2 = x 1 4x2 + x 3
w 3 = x 1 3x2 + x 3
[10 marks]


Problem 3. Find T 1 (w 1; w2; w3) for a given matrix
w 1 = x 1 + 2x2 + x 3
w 2 = 2x1 + x 2 + 4x3
w 3 = 7x1 + 4x2 5x3.
[10 marks]


Problem 4. Let TA : R3 ! R 3
be multiplication by
2 3
4 1 0
A=4 1 3 15
1 1 4
3:
and let e1; e2; and e3 be the standard basis vectors for R
Find the following:
(a) TA (e1) ; TA (e2) ; TA (3e3) ;
(b) T A (e1 + e2 + e3) :
[10 marks]

[Total:40 marks]



–End of assignment –




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6

,MAT2611

ASSIGNMENT 06

Due date: Friday, 27 June 2025



Problem 1

Let T1( x1 ,x2 ,x3) = ( x1 + x2 ,x1 − x2 ,x3) and T2( x1 ,x2 ,x3) = ( x1 ,x2 + x3 ,x3).

(a) Find the standard matrix for T1 and T2.

(b) Find the standard matrix for T1 ◦ T2 and T2 ◦ T1.

(c) Use the matrices obtained in part (b) to find formulas for T1( T2( x1 ,x2 ,x3)) and
T2( T1( x1 ,x2 ,x3)).




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