Linear Algebra 2270
Homework 8
preparation for the quiz on 07/15/2015
Problems:
1. Consider two different bases of R2
1
0
1
0
e = {[ ] , [ ]} , v = {[ ] , [ ]}
0
1
1
2
and two different bases of R3 :
⎡1⎤ ⎡0⎤ ⎡0⎤⎫
⎡1⎤ ⎡0⎤ ⎡0⎤⎫
⎧
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎪
⎪
⎪⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎪
⎪
e = ⎨⎢0⎥ , ⎢1⎥ , ⎢0⎥⎬ , w = ⎨⎢1⎥ , ⎢2⎥ , ⎢1⎥⎬
⎥
⎥
⎥
⎥
⎢
⎥
⎥
⎢
⎢
⎢
⎢
⎢
⎪
⎪
⎪
⎪
⎪
⎪⎢0⎥ ⎢0⎥ ⎢1⎥⎪
⎪
⎪
⎪
⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪
⎭
⎩⎢⎣0⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1⎥⎦⎪
⎭
′
(a) Let T ∶ R3 → R2
⎡ ⎤
⎛⎢⎢x1 ⎥⎥⎞
x + 2x2 + x3
]
T ⎜⎢x2 ⎥⎟ = [ 1
⎥
⎢
x1 − x3
⎝⎢x3 ⎥⎠
⎣ ⎦
Find the following matrices of T in different bases: Me′ ,e (T ), Mw,e (T ), Me′ ,v (T ), Mw,v (T )
(b) Let T ∶ R3 → R3
⎡ ⎤
⎡
⎤
⎛⎢⎢x1 ⎥⎥⎞ ⎢⎢x1 + 2x2 + x3 ⎥⎥
T ⎜⎢x2 ⎥⎟ = ⎢ x1 − x3 ⎥
⎥
⎝⎢⎢x3 ⎥⎥⎠ ⎢⎢
⎥
x2
⎣ ⎦
⎣
⎦
Find the following matrices of T in different bases: Me′ ,e′ (T ), Mw,e′ (T ), Me′ ,w (T ), Mw,w (T )
(c) A matrix I ∈ Rn×n is called an identity matrix, if
where it has 1’s.
⎡1 0 0
⎢
⎢0 1 0
⎢
⎢
I = ⎢0 0 1
⎢
⎢⋮
⋱
⎢
⎢0 0 0
⎣
it has 0’s everywhere apart of the diagonal,
. . . 0⎤⎥
. . . 0⎥⎥
⎥
. . . 0⎥
⎥
0⎥⎥
. . . 1⎥⎦
Consider the identity transformation T ∶ R3 → R3
⎡ ⎤
⎡ ⎤
⎛⎢⎢x1 ⎥⎥⎞ ⎢⎢x1 ⎥⎥
T ⎜⎢x2 ⎥⎟ = ⎢x2 ⎥
⎝⎢⎢x3 ⎥⎥⎠ ⎢⎢x3 ⎥⎥
⎣ ⎦
⎣ ⎦
Find the following matrices of T in different bases: Me′ ,e′ (T ), Mw,e′ (T ), Me′ ,w (T ), Mw,w (T ).
Which of those are 3 × 3 identity matrices?
2. Consider a set of matrices Rm×n with the operation of addition and scalar multiplication, which is
the same as for vectors, but the numbers are arranged in a table, not in a column:
for any c ∈ R:
⎡ a1 1 . . . a1 n ⎤ ⎡ b1 1 . . . b1 n ⎤ ⎡ a1 1 + b1 1 . . . a1 n + b1 n ⎤
⎥ ⎢
⎥
⎢
⎥ ⎢
⎢
⎥ ⎢
⎥ ⎢
⎥
⎢ ⋮
⎥
⋱
⋮ ⎥+⎢ ⋮
⋱
⋮ ⎥=⎢
⋮
⋱
⋮
⎢
⎥ ⎢
⎥ ⎢
⎥
⎢am 1 . . . am n ⎥ ⎢bm 1 . . . bm n ⎥ ⎢am 1 + bm 1 . . . am n + bm n ⎥
⎣
⎦ ⎣
⎦ ⎣
⎦
(1)
⎡ a1 1 . . . a1 n ⎤ ⎡ c⋅a1 1 . . . c ⋅ a1 n ⎤
⎥ ⎢
⎥
⎢
⎢
⎥ ⎢
⎥
⋱
⋮ ⎥=⎢ ⋮
⋱
⋮ ⎥
c⋅⎢ ⋮
⎢
⎥ ⎢
⎥
⎢am 1 . . . am n ⎥ ⎢c ⋅ am 1 . . . c ⋅ am n ⎥
⎣
⎦ ⎣
⎦
(2)
(a) Prove that “⋅” and “+” satisfy V1) – V8), and as a result Rm×n is a vector space. (Hint: the
proof is the same as for vectors, just the numbers are arranged in a table, not in a column)
1
(b) What is the dimension of R2×3 ?
(c) Prove that the set of 2 × 2, symmetric matrices:
a
a
W = {[ 1 1 1 2 ] ∶ a2 1 = a1 2 }
a2 1 a2 2
is a subspace of R2×2 .
(d) Prove that the set of 3 × 3 upper triangular matrices:
⎡a
⎧⎡⎢a1 1 a1 2 a1 3 ⎤⎥⎫
⎫
⎧
a
a ⎤
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎢
⎪ ⎪
⎪⎢⎢ 1 1 1 2 1 3 ⎥⎥
⎥⎪
3×3
∶ a2 1 = a3 1 = a3 2 = 0⎬ = ⎨⎢ 0 a2 2 a2 3 ⎥⎬
W = ⎨⎢a2 1 a2 2 a2 3 ⎥ ∈ R
⎥
⎢
⎥
⎢
⎪
⎪
⎪
⎪
⎪⎢ 0
⎪
⎪⎢a
⎪
⎪
⎪
0 a3 3 ⎥⎦⎪
⎭
⎩⎣
⎭ ⎪
⎩⎣ 3 1 a3 2 a3 3 ⎥⎦
is a subspace of R3×3 .
2