Zero Zeke
Posted in Aircraft Model by admin
on May 9, 2010
Linear Algebra: Linear transformations?
Matrix A =
1 3 2 4
0 6 -4 3
-2 0 8 9
Find a pair of vectors x,y in R^4 that span the set of all x in R^4 that are mapped into the zero vector by the transformation x |----> Ax
I have no idea how to do this problem. The most detailed explanation gets 10 points.
Thanks in advance.
Zeke
This is a question about the kernel of the linear transformation T(x) = Ax. You just have to characterize all of the vectors x in R^4 such that the matrix multiplication with A gives the zero vector in R^3. Let's set it up. Let x = (a,b,c,d). Then Ax = 0 gives
a + 3b + 2c + 4d = 0
0a + 6b -4c + 3d = 0
-2a + 0b + 8c + 9d = 0.
This imposes conditions on the entries a, b, c, and d. The simplest solution method is probably just doing a row reduction on A---get it in reduced row echelon form. rref(A) is
1 0 0 -1
0 1 0 13/12
0 0 1 7/8
This system is equivalent to Ax = 0. It can be interpreted as
a - d = 0
b + (13/12)d = 0
c + (7/8)d = 0
So a = d, b = -(13/12)d, c = -(7/8)d, d = d with no condition on d. Every vector in R^4 that gets mapped to zero is of the form
d(1, -13/12, -7/8, 1)
for any scalar d.
It's not clear to me why you are asked for 2 vectors. Since there is only one variable to play with, namely d, the vectors that get mapped to zero form a 1 dimensional subspace of R^4. But, you can find two. Just take two different values for d. Simple ones would be
(1, -13/12, -7/8,1) (when d =1) and
(24, -26, -21, 24) (when d = 24).
Call one of these x and the other y.
 |
 Tamiya 60318 1 32 Mistubishi A6M5 Zero Fighter Model 52 Zeke $90.50 Time Remaining: 3h 57m Buy It Now for only: $90.50 |
 JO HAN A6M2 ZERO ZEKE OR ZERO RUFE BUILDS TWO WAYS $9.99 Time Remaining: 7h 38m |
 OLD HASEGAWA MITSUBISHI ZERO A6M5 ZEKE IN LARGE 1 32 SCALE VERY NICE $40.00 Time Remaining: 12h 18m |
 Hasegawa 51940 Mitsubishi A6M5 Zero Fighter ZEKE Skeleton Military Model 1 48 $16.28 (5 Bids) Time Remaining: 13h 50m |
 TAMIYA 1 48 61016 MITSUBISHI A6M2 ZERO FIGHTER ZEKE $12.99 Time Remaining: 14h 49m Buy It Now for only: $12.99 |
 Hasegawa 1 48 A6M2b ZERO FIGHTER TYPE 21 ZEKE 09143 $24.56 Time Remaining: 16h 19m Buy It Now for only: $24.56 |
 1 32 TOMY HP 4 1200 MITSUBISHI A6M2 ZERO SEN TYPE 21 ZEKE FIGHTER MOTORIZED $100.00 Time Remaining: 22h 38m Buy It Now for only: $119.99 |
 Mitsubishi A6M2 Zero Fighter ZEKE model plane Tamiya Japan Aircraft Sealed Kit $39.99 Time Remaining: 22h 49m Buy It Now for only: $43.99 |
 TYPE ZERO CARRIER FIGHTER A6M5 ZEKE 148 OTAKI OT2 4 200 MY16P27 $10.95 Time Remaining: 23h |
 SEALED Vintage Mitsubishi ZEKE Japanese ZERO Fighter Plane ARII 1 48 Model Kit $10.00 Time Remaining: 23h 40m |
 Limited Tamiya 1 48 IJN A6M2 ZERO Fighter ZEKE w Kurogane Maintenance Crew $34.90 Time Remaining: 1d 18m Buy It Now for only: $34.90 |
 Tamiya 60318 1 32 Mitsubishi A6M5 Zero Model 52 Zeke $119.99 Time Remaining: 1d 3h 55m Buy It Now for only: $119.99 |
 1962 VINTAGE MONOGRAM 1 48 MITSUBISHI ZERO ZEKE PA73 98 $13.50 (3 Bids) Time Remaining: 1d 23h 23m |
 Tamiya 1 48 Mitsubishi A6M2 Zero Fighter Zeke Propeller Action Series 61509 $25.00 Time Remaining: 1d 23h 42m Buy It Now for only: $40.00 |
 Tamiya 1 48 Mitsubishi A6M5c Zero Fighter Zeke 61027 $20.00 Time Remaining: 1d 23h 42m Buy It Now for only: $30.00 |
 Wing Club L3 Zero Fighter Type 21 Nakajima A6M2b Zeke 1 144 Bandai 042216 $7.99 Time Remaining: 2d 4h 45m |
 Hasegawa D21 A6M2 ZERO FIGHTER TYPE 21 ZEKE 1 72 scale kit $13.80 Time Remaining: 2d 7h 22m Buy It Now for only: $13.80 |
 Hasegawa 1 48 Mitsubishi Zero Type 52 Zeke Comm Flight $25.99 Time Remaining: 2d 10h 48m Buy It Now for only: $25.99 |
 TAMIYA 1 48 A6M2 ZERO FIGHTER ZEKE PLASTIC MODEL KIT $8.00 Time Remaining: 2d 23h 24m |
 Tamiya 60779 1 72 Mitsubishi A6M5 Zero Fighter Zeke Model Kit $18.20 Time Remaining: 3d 3h 15m Buy It Now for only: $18.20 |
 Vintage Tamiya Mitsubishi A6M2 Zero Fighter Zeke 1 48 Scale Airplane Kit New $14.99 Time Remaining: 3d 11h 33m Buy It Now for only: $19.99 |
 TAMIYA 1 48 MITSUBISHI A6M5c ZERO FIGHTER ZEKE $12.00 Time Remaining: 3d 13h 8m |
 1 32 MITSUBISHI TYPE ZERO ZEKE A6 M5 HASEGAWA VINTAGE NEW $59.99 Time Remaining: 3d 21h 12m Buy It Now for only: $59.99 |
 HASEGAWA MINI CRAFT MITSUBISHI TYPE ZERO A6M5 ZEKE 1 32 SCALE $27.99 Time Remaining: 3d 22h 4m |
 Tamiya 60317 1 32 WW2 IJN Japanese Navy A6M2b ZERO Zeke Type 21 Fighter Kit $78.90 Time Remaining: 4d 6h 13m Buy It Now for only: $78.90 |
 Bandai Wing Club L3 Zero Fighter Type 21 Nakajima A6M2b 1 144 Zeke 040501 $9.99 Time Remaining: 4d 7h 33m |
 Bandai Wing Club L3 Zero Fighter Type 21 Nakajima A6M2b 1 144 Zeke 040501 $7.99 Time Remaining: 4d 7h 36m |
 Tamiya Japanese Zero Mitsubishi A6M2 FIGHTER ZEKE 148 SCALE $14.00 Time Remaining: 4d 16h 12m |
 Rare vintage WW2 LS PLASTIC MODEL KIT A6M5 ZERO ZEKE 1 72 $2.00 Time Remaining: 4d 21h 12m |
 Hasegawa 1 48 Mitsubishi A6M5 Zero Fighter Zeke Type52 $14.95 Time Remaining: 4d 21h 22m Buy It Now for only: $14.95 |
 JO HAN 172 Scale A6M2 ZERO ZEKE or RUFE A 105 $0.99 (1 Bid) Time Remaining: 4d 22h 15m |
 1 48 Hasegawa Mitsubishi A6M5 Zero Fighter Zeke Type 52 $15.00 Time Remaining: 4d 23h 8m |
 Tamiya Mitsubishi A6M2 Zero Fighter ZEKE 1 48 Scale $9.99 Time Remaining: 5d |
 NMIB OTAKI JAPANESE ZERO A6M5 ZEKE 1 48 PLASTIC AIRPLANE MODEL KIT 1970S $14.99 Time Remaining: 5d 16m Buy It Now for only: $25.00 |
 Gakken 12 ZeroZeke Fighter of WW2 $24.98 Time Remaining: 5d 11h 19m Buy It Now for only: $24.98 |
 ZERO FIGHTER A6M5 Zeke Type 52 Mitsubishi Hasegawa 1 48 $0.11 (2 Bids) Time Remaining: 5d 18h 24m |
 Mitsubishi A6M2b Zero Fighter Zeke Type 21 172 Hasegawa 00451 MY21P13 $9.95 Time Remaining: 5d 23h 1m |
 Tamiya 60317 1 32 Mistubishi A6M2b Zero Fighter 21 Zeke Model Kit FREESHIP $114.90 Time Remaining: 6d 1h 49m Buy It Now for only: $114.90 |
 TAM60317 A6M2b Model 21 Zeke Zero Fighter 1 32 Tamiya $127.50 Time Remaining: 6d 12h 46m Buy It Now for only: $127.50 |
 TAM60318 A6M5 Model 52 Zeke Zero Fighter 1 32 Tamiya $127.50 Time Remaining: 6d 12h 46m Buy It Now for only: $127.50 |
 TAM61016 A6M2 Type 21 Zero Fighter Zeke 1 48 Tamiya $12.75 Time Remaining: 6d 12h 47m Buy It Now for only: $12.75 |
 TAM61027 Mitsubishi A6M5c Zero Fighter Zeke 1 48 Tamiya $13.60 Time Remaining: 6d 12h 47m Buy It Now for only: $13.60 |
 TAM61103 Mitsubishi A6M5 5a Zero Zeke Fighter 1 48 Ta $48.45 Time Remaining: 6d 12h 48m Buy It Now for only: $48.45 |
 Tamiya 1 32 Mitsubishi A6M2B Zero Zeke Model Kit 60317 TAM60317 $109.99 Time Remaining: 6d 13h 48m Buy It Now for only: $109.99 |
 TOMY JAPANESE NAVY FIGHTER MITSUBISHI A6M2 ZERO SEN ZEKE AIRPLANET MODEL KIT $4.99 (1 Bid) Time Remaining: 6d 18h |
 Tamiya 1 48 A6M5 5A Zero Fighter Zeke Model Kit 61103 TAM61103 $42.75 Time Remaining: 6d 18h 31m Buy It Now for only: $42.75 |
 NEW Hasegawa 1 48 Mitsubishi A6M5 Zero Fighter 52 Zeke 09072 NIB $36.99 Time Remaining: 6d 20h 30m Buy It Now for only: $36.99 |
 NEW Tamiya 1 48 Mitsubishi A6M5 5a Zero Fighter Zeke 61103 NIB $39.99 Time Remaining: 6d 20h 42m Buy It Now for only: $39.99 |
 Tamiya 60318 Mitsubishi A6M5 Zero Fighter Model 52 Zeke $122.00 Time Remaining: 8d 6h 18m Buy It Now for only: $122.00 |
 Tamiya 60317 Mitsubishi A6M2b Zero Fighter Model 21 Zeke $122.00 Time Remaining: 8d 6h 18m Buy It Now for only: $122.00 |
 Mitsubishi A6M5c Zero Fighter Zeke 1 48 Tamiya MIB Kit no 61027 Nice $19.95 Time Remaining: 9d 1h 54m Buy It Now for only: $19.95 |
 JO HAN 1 72 MITSUBISHI A6M2 ZERO ZEKE OR RUFE A 105 CARRIER FIGHTER FLOATPLANE $15.75 Time Remaining: 9d 3h 26m Buy It Now for only: $15.75 |
 Tamiya 1 48 61108 Mitsubishi A6M3 3a Zero Fighter ZEKE $34.99 Time Remaining: 9d 12h 10m Buy It Now for only: $34.99 |
 Tamiya 61103 Mitsubishi A6M5 5a Zero Fighter Mdeol 52 Zeke $41.00 Time Remaining: 9d 21h 2m Buy It Now for only: $41.00 |
 Tamiya 61108 Mitsubishi A6M3 3a Zero Fighter Mdeol 22 Zeke $41.00 Time Remaining: 9d 21h 2m Buy It Now for only: $41.00 |
 Tamiya 1 32 WW2 IJN A6M5 ZERO Zeke Type 52 Fighter NIB $89.00 Time Remaining: 10d 3h 57m Buy It Now for only: $89.00 |
 Tamiya 61108 Mitsubishi A6M3 3a Zero Fighter ZEKE 1 48 $32.50 Time Remaining: 10d 4h 33m Buy It Now for only: $32.50 |
 HASEGAWA 1 72 WWII JAPANESE MITSUBISHI A6M3 ZERO SEN TYPE 22 ZEKE JS 076200 $15.75 Time Remaining: 10d 5h 11m Buy It Now for only: $15.75 |
 Tamiya 61108 1 48 Mitsubishi A6M3 3a Zero Model 22 Zeke $39.99 Time Remaining: 11d 21h 23m Buy It Now for only: $39.99 |
 1 48 Mitsubishi A6M2 Zero Zeke Tamiya 61016 $13.75 Time Remaining: 12d 16h 29m Buy It Now for only: $13.75 |
 Tamiya 61103 Mitsubishi A6M5 5a Zero Zeke Fighte 148 $51.24 Time Remaining: 12d 17h 21m Buy It Now for only: $51.24 |
 Tamiya 61108 Mitsubishi A6M3 3a Zero Fighter ZEKE 148 $48.74 Time Remaining: 12d 17h 58m Buy It Now for only: $48.74 |
 Hasegawa 452 1 72 Mitsubishi A6M5 Zero Fighter Type 52 Zeke $14.99 Time Remaining: 12d 18h 21m Buy It Now for only: $14.99 |
 Hasegawa 451 1 72 Mitsubishi A6M2 Zero Fighter Type 21 Zeke $14.99 Time Remaining: 12d 18h 21m Buy It Now for only: $14.99 |
 FIGURE MITSUBISHI ZERO FIGHTER PLANE Model 32 A6M3 Zeke $24.99 Time Remaining: 14d 3h 23m Buy It Now for only: $24.99 |
 Fine Scale Modeler July 92 A6M2 Zero Zeke Crusader F 8E Woodcarving USS Oregon $12.95 Time Remaining: 14d 5h 4m Buy It Now for only: $12.95 |
 Tamiya 60779 Mitsubishi A6M5 Zero Fighter ZEKE 1 72 scale kit $16.80 Time Remaining: 14d 7h 41m Buy It Now for only: $16.80 |
 1 144 Mitsubishi A6M2 Zero Zeke Academy Minicraft OOP $14.95 Time Remaining: 14d 16h 53m Buy It Now for only: $14.95 |
 TAMIYA 1 48 JAPANESE A6M5C MODEL 52 ZEKE ZERO FIGHTER $17.97 Time Remaining: 14d 17h 23m Buy It Now for only: $17.97 |
 TAMIYA 1 72 Mitsubishi A6M5 Zero Fighter Zeke 60779 $20.15 Time Remaining: 14d 20h 27m Buy It Now for only: $20.15 |
 Hasegawa JT43 MITSUBISHI A6M2b ZERO TYPE 21 ZEKE 1 48 scale kit $26.80 Time Remaining: 15d 5h 40m Buy It Now for only: $26.80 |
 OTAKI Model ZERO CARRIER FIGHTER A6M5 ZEKE Kit OT 2 4 200 $18.00 Time Remaining: 15d 19h 3m Buy It Now for only: $18.00 |
 Tamiya 1 48 Mitsubishi A6M5c Zero Fighter Zeke Kit $16.99 Time Remaining: 16d 42m Buy It Now for only: $16.99 |
 Mitsubishi A6M2 Zero Fighter Zeke Tamiya 1 48 scale $15.00 Time Remaining: 16d 17h 2m Buy It Now for only: $15.00 |
 NEW Tamiya 1 72 Mitsubishi A6M5 Zero Fighter Zeke 60779 NIB $19.99 Time Remaining: 16d 17h 14m Buy It Now for only: $19.99 |
 Vintage Monogram MITSUBISHI ZERO A6M5 ZEKE Japanese Fighter 1 48 SEALED KIT 6799 $22.95 Time Remaining: 16d 17h 42m Buy It Now for only: $22.95 |