Let "V" and "W" be finite dimensional vector spaces and
T: V-->W be linear.
a)prove that if dim (V) < dim(W), then "T" cannot be onto.
b) Prove that if dim(V) > dim(W), then T cannot be one one.
2007-09-20 09:13:47 · 4 answers · asked by amjad 1 in Science & Mathematics ➔ Mathematics
Just see it.
a) some w's not hit, not onto.
b) some w's hit several times, not 1-1.
2007-09-20 09:26:00 · answer #1 · answered by Anonymous · 0⤊ 0⤋
You got A right.. B - [cos(u) , -sin(u)] [sin(u) , cos(u) ] Its late so i cant be bothered to explain this one C- [0 , 1] [1 , 0] Yhis is simple as reflection in that axes is just swappin the x,y co-ordinates D- [1 , 0] [0 , 0] disregarding the y compnent so just projecting onto x-axis. Similar idea for prjection onto y axis [0 , 0] [0 , 1]
2016-05-19 04:35:49 · answer #2 · answered by ranae 3 · 0⤊ 0⤋
Pictures worth a thousand proofs, see
http://mathworld.wolfram.com/Surjection.html
2007-09-20 10:01:13 · answer #3 · answered by ? 5 · 0⤊ 0⤋
They both fall out (trivially) from the definition(s) of 1-1 and onto.
Doug
2007-09-20 09:25:30 · answer #4 · answered by doug_donaghue 7 · 0⤊ 0⤋