Study Guide
Block 3 : S e l e c t e d Topics i n L i n e a r Algebra
Unit 4: L i n e a r Transformations
1. Overview
W e have a l r e a d y s e e n many i n s t a n c e s i n o u r c o u r s e where t h e
concept of a l i n e a r f u n c t i o n was most c r u c i a l .
I t turns out
t h a t t h e g e n e r a l concept of a l i n e a r t r a n s f o r m a t i o n i s b e s t
handled i n t e r m s of viewing them a s s p e c i a l mappings o f v e c t o r
s p a c e s i n t o v e c t o r s p a c e s . Thus, t h e aim o f t h i s u n i t i s t o
show how t h i s s t u d y i s handled, and i t i s o u r hope t h a t s e e i n g
t h e g e n e r a l s t r u c t u r e w i l l make it c l e a r a s t o what common
p r o p e r t i e s a r e s h a r e d by a l l l i n e a r t r a n s f o r m a t i o n s .
Study Guide
Block 3: S e l e c t e d T o p i c s i n L i n e a r Algebra
U n i t 4: L i n e a r T r a n s f o r m a t i o n s
2.
L e c t u r e 3.040
-
0
Study Guide
Block 3 : S e l e c t e d Topics i n L i n e a r Algebra
Unit 4: L i n e a r T r a n s f o r m a t i o n s
3.
Exercises:
Let V
[u1.u21
=
and l e t f:V
+
+ x2u2 , compute f (xl,x2)
a.
L e t t i n g (xl , x 2 ) d e n o t e xlul
b.
With f a s above, l e t vl = 7ul
f (vl),f ( v 2 ) , and f (vl
f
c.
+
V b e t h e l i n e a r f u n c t i o n d e f i n e d by
+
5u2 and v2 = 2ul
v 2 ) ; and show t h a t f (vl
+
+
.
3u2.
Compute
v 2 ) = f (vl)
+
(v2)
:
t
and u2 w i t h 3 , d e s c r i b e f i n terms o f how
with
1
it maps t h e xy-plane o n t o t h e uv-plane.
Identifying u
L e t V = [u1,u2,u3] ; and l e t al = ( 1 , 2 , 3 ) , a 2 = ( 4 , 5 , 6 ) ,
a 3 = ( 7 , 8 , 9 ) EV. Suppose T:V + w i s l i n e a r where W = [wl ,w2,w3 , w 4 ]
a.
Is it p o s s i b l e t h a t T ( a l ) = ( 3 , 1 , 2 , 4 ) , ~ ( a =~ ( 4
) , 2 , 1 , 5 ) and
T(a3) = ( 2 , 3 , 4 , 1 ) ?
b.
Explain.
L e t yl = (1,1,1),y2 = ( 1 , 2 , 3 ) t y3 = ( 2 , 3 1 5 ) I and Y 4 = ( 3 1 7 1 6 )
Express T ( y 4 ) a s a l i n e a r combination of T (yl) I T ( y 2 ) t and T ( y 3 )
Define t h e l i n e a r t r a n s f o r m a t i o n f : V
V, where V = [ u l I U 2 l I by
f ( u l ) = -3ul + 2u2 and f ( u 2 ) = 4u, - u2.
+
a.
Letting
use t h e method d e s c r i b e d i n t h e l e c t u r e t o e x p r e s s f ( v ) =
f (xlul
+
x 2u 2 ) a s a p r o d u c t of m a t r i c e s .
( c o n t i n u e d on n e x t page)
.
Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
Unit 4: Linear Transformations
3.4.3 (L) c o n t i n u e d
b.
D o t h e same a s i n ( a ) b u t now u s e t h e m a t r i x
L e t V = [ul,u2, u31 and l e t t h e l i n e a r t r a n s f o r m a t i o n f : V
+
V be
d e f i n e d by
(ul) = u 1 + u 2 + u3 f ( u 2 ) = 2u + 3u2 + 3u3 1
f f u ) = 3ul + 4u2 + 6u3
3
.
Now, l e t v = x u + x2u2
11
+
f
x3u3.
a.
Compute f ( v ) w i t h o u t t h e u s e of m a t r i c e s .
b.
Compute f ( v ) u s i n g t h e m a t r i c e s
and
c.
U s e BT and
;)XT t o compute f ( v ) i n terms o f
L e t V = [ u l r u 2 ] and l e t f b e t h e l i n e a r t r a n s f o r m a t i o n f:V + V
d e f i n e d by f (ul) = u1 + 2u2 , f ( u 2 ) = 3ul + 5u2. L e t vl = u l + u 2
and v2 = 2ul
(
+
u2.
c o n t i n u e d on n e x t page 1
Study Guide
Block 3: S e l e c t e d T o p i c s i n L i n e a r A l g e b r a
U n i t 4: L i n e a r T r a n s f o r m a t i o n s
3.4.5 (L) c o n t i n u e d
a . Show t h a t V = [vl,v2]
and e x p r e s s ul and u2 i n t e r m s of vl and v 2 .
U s e t h i s r e s u l t t o e x p r e s s f ( v l ) and f ( v 2 ) a s l i n e a r combinations
o f v1 and v 2 .
What i s t h e m a t r i x of c o e f f i c i e n t s o f f r e l a t i v e
t o t h e b a s i s {v1,v2} ?
b. L e t v = 4ul
+
7u2.
E x p r e s s f ( v ) a s a l i n e a r combination of ul
and u2 and a l s o a s a l i n e a r combination of v
1
c. Suppose V = [ a i , a 2 1 and t h a t a l s o V = [B1, B 2 1
Suppose a l s o t h a t T:V
+
and v 2 .
.
Say
V is the l i n e a r transformation defined
by
show t h a t t h e m a t r i x BAB-I
represents T r e l a t i v e t o the basis
3.4.6
(optional)
a.
Show t h a t i f X - ~ A X = I , t h e n A = I.
b.
Show t h a t i f X - ~ A X = 0, t h e n A = 0.
Let
A =
( c o n t i n u e d on n e x t page)
Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
Unit 4: L i n e a r T r a n s f o r m a t i o n s
-
-
3.4.7
-
-
continued
be t h e matrix of c o e f f i c i e n t s of t h e l i n e a r transformation
f:V
+
.
V r e l a t i v e t o t h e b a s i s {u1,u2,u31
Now, l e t vl = u
1
+
+ 5u2 + 6u3, and v3 = 3ul + 6u2 + 10u3. Show
1
t h a t V = [v1,v2,v31, and u s e t h e method d e s c r i b e d i n E x e r c i s e
2u2
+
8.4.5
3u3, v2 = 2u
t o express t h e matrix of c o e f f i c i e n t s of f r e l a t i v e t o
t h e b a s i s {v1,v2,v3)
.
3.4.8(L)
L e t V = [ul ,u2 ,u31 and l e t f :V
+
V be t h e l i n e a r t r a n s f o r m a t i o n
d e f i n e d by
D e s c r i b e t h e s p a c e f(V) and show t h a t i t s dimension i s 2.
Also,
d e s c r i b e t h e n u l l s p a c e of V w i t h r e s p e c t t o f .
3 .4.9
(optional)
[This i s a g e n e r a l i z a t i o n of t h e p r e v i o u s e x e r c i s e . ]
L e t V = [v1,v2,v3,v4] and W = [wlrw21.
l i n e a r t r a n s f o r m a t i o n d e f i n e d by
Suppose f:V
+
W is the
a. Show t h a t f ( v ) = W.
I n p a r t i c u l a r , f i n d a l and a2€ V such t h a t
Also, f i n d a b a s i s f o r N f .
£ ( a l ) = w1 and £ ( a 2 ) = w2.
b. Find a row-reduced b a s i s f o r N f and show how x3and x 4 must b e
r e l a t e d t o xl and x2 i f ( x l r x 2 , x 3 , x 4 ) ~ N f .
c. Find a l l vsV s u c h t h a t f ( v ) = 5wl
+
6w2*
Study Guide
Block 3: S e l e c t e d Topics i n L i n e a r Algebra
U n i t 4: L i n e a r T r a n s f o r m a t i o n s
3.4.10
(optional)
[This e x e r c i s e i s n o t c r u c i a l h e r e b u t it i s v e r y i m p o r t a n t i n
Unit 6.1
L e t V b e a v e c t o r s p a c e and l e t c b e a f i x e d r e a l number.
f:V
+
V is linear.
Define w = { v E v : ~ ( v )= c v
a subspace of V and, moreover, t h a t f (w) cW.
1.
Suppose
Prove t h a t w i s
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Resource: Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra
Prof. Herbert Gross
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