Which of the transformations are onto?(adsbygoogle = window.adsbygoogle || []).push({});

1) T:R^{2}-> R^{2}, where T(x,y) = (5x-y, 0)

I don't know if I'm understanding this correctly but this transformation is NOT onto because if I let

5x-y = a

0 = b

this means that b doesn't cover all the range of T? Could someone explain it better if I'm wrong.

2) T:R^{3}-> R^{2}where T(x,y,z) = (x+y, x-z)

so I equate this to

x+y = a

x-z = b

which makes the matrix

row1 = (1 1 0 a)

row 2 = (1 0 -1 b)

then once I reduce it to row echelon form, I ultimately get

row 1 = (1 0 -1 b)

row 2 =( 0 1 1 a-b)

then I get stuck because I don't understand what that means..I'm going to guess it's NOT onto because for any value a and b, I can't get and x,y, or z?

Please correct me if I'm wrong.

3) T:R^{2}-> R^{3}, where T(x,y) = (y, x, x-y)

I equate this again to

y = a

x= b

x-y = c

which forms the matrix

row 1 = (0 1 a)

row 2 =(1 0 b)

row 3 = (1 -1 c)

then perform row echelon to ultimately get

row 1 = (1 0 b)

row 2 = (0 1 a)

row 3 = (0 -1 -b+c)

I think this is NOT onto again because of the last row but I can't be 100% sure. Any guidance would help a lot.

I know this is much to ask but if you can even help me with just one, it would mean a lot. Thank you for any help.

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# Linear transformation being onto or not (Three problems I need help understanding)

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