MATH140 New York University Linear Algebra Problems Homework 10

There are 6 questions in this homework assignment, and I will also provide the lecture notes for this homework.

Linear Algebra (MATH140/MA-UY3044)
Fall 2019
HW10: sections 7.1, 7.2, 8.1, 8.2 (lectures 20 – 22)
(Due Monday 12/9/2019 before midnight)


1 1 −1
1. (12 points) Let A =
.
1 1 −1
(a) (8 points) Find the singular value decomposition, A = U ΣV T .
(b) (4 points) Based on your answer to part (a), write an orthonormal basis for each
of the four fundamental subspaces of A.
2. (9 points/3 each) Prove that T is a linear transformation.
(a) T : R3 → R2 , where T (x1 , x2 , x3 ) = (x1 − x2 , 2×3 ).
(b) T : R2 → R3 , where T (x1 , x2 ) = (x1 + x2 , 0, 2×1 − x2 ).
(c) T : P2 → P3 , where T [f (x)] = xf (x) + f 0 (x).
3. (6 points/2 each) Let T : R2 → R2 . State why T is not linear.
(a) T (x1 , x2 ) = (1, x1 ).
(b) T (x1 , x2 ) = (x1 , x21 ).
(c) T (x1 , x2 ) = (1 + x1 , x2 ).
Linear Algebra (MATH140/MA-UY3044)
Fall 2019
4. (3 points/1 each) Let β and β̃ be the standard bases for Rn and Rm , respectively. Find
the matrix A associated with the linear transformation T : Rn → Rm in the following
cases:
(a) T : R2 → R3 , where T (x1 , x2 ) = (2×1 − x2 , 3×1 + 4×2 , x1 ).
(b) T : R3 → R2 , where T (x1 , x2 , x3 ) = (2×1 + 3×2 − x3 , x1 + x3 ).
(c) T : R3 → R, where T (x1 , x2 , x3 ) = 2×1 + x2 − 3×3 .
5. (10 points) Let T : R2 → R3 , where
(x1, x2) =
(x1 − x2 , x1 , 2×1 + x2 ). Let β be the
 T 

0
2 
 1
standard basis for R2 and β̃ = 1 , 1 , 2 be a basis for R3 . Find the matrix


0
1
3
A associated with the linear transformation T .
   
1
2
If the basis for R had been β =
,
intead of the standard basis, what would
2
3
the matrix associated with the transformation be?
2
6. (10 points/5 each) In each of the following cases, find the change of basis matrix that
changes β 0 -coordinates into β-coordinates.
   
   
−4
2
2
−4
0
(a) β =
,
,β =
,
, where β and β 0 are bases for R2 .
3
−1
1
1
(b) β = {x2 − x + 1, x + 1, x2 + 1}, β 0 = {x2 + x + 4, 4×2 − 3x + 2, 2×2 + 3}, where
β and β 0 are bases for P2 .