The map (1 4 -2 3 12 -6 ) is not surjective Let's understand the difference between these two examples: General Fact Let A be a matrix and let Ared be the row
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AB = I into an explicit bijective proof of the identity BA = I Letting A and B be the Kostka matrix and its inverse, this settles an open problem posed by E˘gecio˘glu
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Bijective matrix algebra - ScienceDirectcom
AB = I into an explicit bijective proof of the identity BA = I Letting A and B be the Kostka matrix and its inverse, this settles an open problem posed by E˘gecio˘glu
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If it is invertible, give the inverse map 1 The linear mapping R3 → R3 which scales every vector by 2 Solution note: This is surjective, injective, and invertble
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The map (1 4 -2 3 12 -6 ) is not surjective Let's understand the difference between these two examples: General Fact Let A be a matrix and let Ared be the row
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INJECTIVE, SURJECTIVE AND INVERTIBLE
DAVID SPEYER
Surjectivity: Maps which hit every value in the target space Let's start with a puzzle. I have a remote control car, controlled by 3 buttons. When I hold down the red button, it moves in direction1 2 ; when I hold down the green button it moves in direction 2 3 ; when I hold down the blue button, it moves in direction3 5 . Can I get anywhere in the plane? For example, can I get to 12 19 Letrbe the amount of time I hold down the red button, and writegandbfor the green and blue buttons. So we move1 25 2 33 0 @r g b1 A and we want to solve 1 23 2 35 0 @r g b1 A =12 19We run through the usual row reduction process
1 23 01 1 0 @r g b1 A =12 5 1 23 0 11 0 @r g b1 A =12 5 1 01 0 11 0 @r g b1 A =2 5 So there are lots of solutions, the simplest of which is to hold down the red button for 2 seconds and the green for 5.Notice that, if I wanted to move to5
2 instead, I only need to redo the computations on the right hand side of the equations; the left hand sides stay the same.Foranyx
y , we can nd1(r;g;b) values which will move us in directionx
y . There is a term for this: Vocabulary.A linear mapA:Rk!R`is calledsurjectiveif, for everyvinR`, we can nduin R kwithA(u) =v.1From the physical motivation from this problem, in only makes sense to look at solutions wherer,gandb0.
In fact, such solutions exist in this case. The subject of solving linear equations together with inequalities is studied
in Math 561. I'll ignore this issue. 12 DAVID SPEYER
Another word which is sometimes used isonto.
So we say that1 23
2 35 is surjective. Let's say a new car comes on the market. It moves by the matrix 1 42 3 126 . Can we still go anywhere? Let's run the row reduction algorithm again. We want to move to positionx y 1 42 3 126 0 @r g b1 A =x y 1 42 0 0 0 0 @r g b1 A =x y3xSo we can only go to
x y ify3x= 0. In other words, this car can only drive along the line y= 3x.The map1 42
3 126 isnotsurjective. Let's understand the dierence between these two examples: General Fact.LetAbe a matrix and letAredbe the row reduced form ofA. IfAredhas a leading1in every row, thenAis surjective. IfAredhas an all zero row, thenAis not surjective.
Remember that, in a row reduced matrix, every row either has a leading 1, or is all zeroes, so one of these two cases occurs.Injectivity: Maps that don't destroy information
Wanda owns two types of pets: birds and cats. I ask her how many of each she has. She replies: \My pets have 14 legs, 10 eyes and 5 tails." Can we gure out how many of each animal there are? Letbbe the number of birds andcthe number of cats. So0 @2 4 2 2 1 11 Ab c =0 @14 10 51A
We proceed as usual
0 @1 2 2 2 1 11 Ab c =0 @7 10 51A 0 @1 2 02 011 Ab c =0 @7 4 21
A 0 @1 2 0 1 011 Ab c =0 @7 2 21
A 0 @1 0 0 1 0 01 Ab c =0 @3 2 01 A
So there are 3 birds and 2 cats.
INJECTIVE, SURJECTIVE AND INVERTIBLE 3
Yes, Wanda has given us enough clues to recover the data.On the other hand, suppose Wanda said
\My pets have 5 heads, 10 eyes and 5 tails."Then we get
0 @1 1 2 2 1 11 Ab c =0 @5 10 51A 0 @1 1 0 0 0 01 Ab c =0 @5 0 01 A: All we can conclude is that the total number of pets is 5; we can't tell how many are cats and how many are birds. Wanda has wickedly failed to give us enough information! Vocabulary.A linear mapA:Rk!R`is calledinjectiveif, for everyvinR`, there is at most oneuinRkwithA(u) =v. In other words,Adoes preserves enough data to recoveru. Another word which is sometimes used isone to one. So0 @2 4 2 2 1 11 A is injective but0 @1 1 2 2 1 11 A is not. We have discussed before how, if there are columns without leading 1's, they give us multiple solutions. If, on the other hand, every column has a leading 1 in it, then there is at most one solution. In our new language: General Fact.LetAbe a matrix and letAredbe the row reduced form ofA. IfAredhas a leading