T(x,y,z,w)=(3y−4z,x+2y+4z−w,7z,−y−w)
Write the standard matrix for T.
A=⎣⎡0100320−1−44700−10−1⎦⎤One-to-one is the same as onto for square matrices.
In general, a transformation T is both one-to-one and onto if and only if T(x)=bhas exactly one solution for all b in Rm.
⎣⎡0100320−1−44700−10−10000⎦⎤ Swap rows 1 and 2:
⎣⎡1000230−14−470−100−10000⎦⎤ R2=R2/3
⎣⎡1000210−14−4/370−100−10000⎦⎤ R1=R1−(2)R2
⎣⎡1000010−120/3−4/370−100−10000⎦⎤ R4=R4+R2
⎣⎡1000010020/3−4/37−4/3−100−10000⎦⎤ R3=R3/7
⎣⎡1000010020/3−4/31−4/3−100−10000⎦⎤ R1=R1−(20/3)R3
⎣⎡100001000−4/31−4/3−100−10000⎦⎤ R2=R2+(4/3)R3
⎣⎡10000100001−4/3−100−10000⎦⎤ R4=R4+(4/3)R3
⎣⎡100001000010−100−10000⎦⎤ R1=R1−R4
⎣⎡100001000010000−10000⎦⎤ R4=R4⋅(−1)
⎣⎡10000100001000010000⎦⎤The columns of matrix are linearly independent, which happens precisely when the matrix has a pivot position in every column.
Therefore T is one-to-one.
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