Linear Algebra 2270
Homework 7
preparation for the quiz on 07/08/2015
Problems:
1. Is there a linear transformation T ∶ R2 → R2 such that the following is satisfied?
1
2
0
−2
1
5
T ([ ]) = [ ] , T ([ ]) = [ ] , T ([ ]) = [ ]
0
1
1
−1
2
0
2. Find the composition T2 ○ T1 of the following linear transformations:
⎡x1 + x2 ⎤
⎥
⎢
x
⎥
⎢
1
T1 ∶ R2 → R3 , T1 ([ ]) = ⎢ −2x1 ⎥
⎥
⎢
x2
⎢x1 − x2 ⎥
⎦
⎣
⎡ ⎤
⎛⎢⎢y1 ⎥⎥⎞
T2 ∶ R → R, T2 ⎜⎢y2 ⎥⎟ = [y1 + y2 − 2y3 ]
⎝⎢⎢y3 ⎥⎥⎠
⎣ ⎦
3
3. Find the inverse of the following linear transformations. Using the definition, show that the inverses
are linear.
(a)
x
x + x2
T ∶ R2 → R2 , T ([ 1 ]) = [ 1
]
x2
−2x1
(b)
x
x − 2x2
T ∶ R2 → R2 , T ([ 1 ]) = [ 1
]
x2
2x1 − 4x2
Hint: (b) is tricky...
4. Let W ≺ R3 be defined as:
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
{⎢x2 ⎥ ∶ x1 + x2 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
dim(W ) = 2, so W ∼ R2 .
(a) Find a linear transformation T1 ∶ W → R2 that is 1-1 and onto.
(b) Find a linear transformation T2 ∶ R2 → W that is 1-1 and onto.
5. (extra problem, will not be on the quiz) Let V be a vector space. W say that V is a direct sum
of W1 , W2 if W1 ≺ V, W2 ≺ V and for any x ∈ V there are unique y1 ∈ W1 , y2 ∈ W2 such that
x = y1 + y2
We denote it by
V = W1 ⊕ W2
1
(a) Show that
V = W1 ⊕ W2 ⇒ W1 ∩ W2 = {0}
(b) Let V = R3 .
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
W1 = {⎢x2 ⎥ ∶ x1 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
W2 = {⎢x2 ⎥ ∶ x2 = 0, x3 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
Show that V = W1 ⊕ W2
(c) Let V = R3 .
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
W1 = {⎢x2 ⎥ ∶ x1 − x2 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
⎡x1 ⎤
⎢ ⎥
⎢ ⎥
W2 = {⎢x2 ⎥ ∶ x2 = 0, x3 = 0}
⎢ ⎥
⎢x3 ⎥
⎣ ⎦
Show that V = W1 ⊕ W2
(d) Let dim(V ) = n < ∞. Let v = {v1 , . . . , vn } be a basis of V . Let m be any integer larger than 0
and less than n. Define sub spaces W1 , W2 as
W1 = span(v1 , . . . , vm )
W2 = span(vm+1 , . . . , vn )
Show that V = W1 ⊕ W2 .
(e) Let V = W1 ⊕ W2 and dim(V ) = n < ∞. Show that there is a basis v = {v1 , . . . , vn } and an
integer m such that
W1 = span(v1 , . . . , vm )
W2 = span(vm+1 , . . . , vn )
Hint: Notice that W1 , W2 has to be finite dimensional as subspaces of a finite dimensional
space V . Take a basis of W1 and a basis of W2 . Show that if you take all of the vectors from
those two bases, they form a basis of V .
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