Question

Find a matrix A that completely determines the function T(x, y) = (2y − 3x, x − 4y, 0, x).
Determine if T is one-to-one and onto.

Answer 1

Salution we have to find a matx A +hat Com determines ne funetfon T Cy-3x,x-47,0,x) Let us ftnd the matodx ouprasentation T with Stes beat to the Standond basis i-e. u0),0 to standand basis i-e. 7C,0,0,0),co,1,0,0),10,00)(0,0,0,13 TI0)-3,1,0,1) = -3(1,0,0,0)+1C0,1,0,0) +0c0,0,1,0)O,0,0,1 T(0,)-2,-4,0,0) 2U,00,0)+t-u) to,170,0) t 0co,0,,0)0C90,0,1) ritngeoluma oine coefticients ee,e1,egfey then A- -4 Juechively ux2 Noo, uwe have to check oheathen T is one-ona and onto we know hat is nullitynLT)- o hen T is One One and onto dimension e null 6pace et TX) nullit nT) e. TCy)o Cy-3x,x-"j,o,x)= (0,0,0,0) 2y-3x O Sc-uyo O O om these we obtalned null space > no) = o 's రెరం I is one-one and onto