Can a P2 to P3 linear transformation matrix be a onto?

Can a P2 to P3 linear transformation matrix be a onto? I ask as I got the P2 matrix into reduced row ehleon and found all rows to be independant (leading 1s). This everything from P2 maps onto P3? ( I cannot give deaild, typing from phone, thanks for help!)

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  • 5 months ago

    If Pn is the space of all polynomials of degree at most n, then it can't happen. P2 is 3-dimensional and P3 is 4-dimensional, so the transformation matrix has 4 rows and 3 columns. Every nonzero row in a reduced row echelon matrix has it's leading 1 as the only nonzero element in its column. So you can't have more nonzero rows than columns. It's impossible for all 4 rows to be nonzero.

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