For the following transformation T either give its standard matrix (i.e. the mat

Question

For the following transformation T either

• give its standard matrix (i.e. the matrix relative to thestandard bases of the domain of T and the codomain of T), if T islinear, or

T ?
ç
ç
ç
è é
ê
ê
ê
ë x y z ù
?
?
?
û ?
÷
÷
÷
? = é
ê
ê
ë 6 x-y+5 z -6 x-y+z+2 ù
?
?
û Enter your answer as follows. Take care to read theseinstructions; the usual type/syntax checking is not possible with"matrix" entries. So type/syntax errors will attractpenalties.

• If T is linear enter the entries ofits standard matrix starting at the cell in the top lefthandcorner, and leave any cells you don't needempty.
• Similarly, to enter a counter-example of L2enter a scalar k and three vectors
  k   v    T(kv)     k T(v) in that order in the 4 cells of the top row of thetable, leaving all other cells empty. To be acounter-example of L2, you musthave
  T(k v) ¹ k T(v) Shorthand notation for vectors.   Usethe following shorthand to enter your vectors:

  separate the entries by commas and enclose inc( ), e.g. c(1, 2 ,3) representsthe column vector
  é
  ê
  ê
  ê
  ë 1 2 3 ù
  ?
  ?
  ?
  û

Answer: Enter your answer as follows. Take care to read theseinstructions; the usual type/syntax checking is not possible with"matrix" entries. So type/syntax errors will attractpenalties.

• If T is linear enter the entries ofits standard matrix starting at the cell in the top lefthandcorner, and leave any cells you don't needempty.
• Similarly, to enter a counter-example of L2enter a scalar k and three vectors
  k   v    T(kv)     k T(v) in that order in the 4 cells of the top row of thetable, leaving all other cells empty. To be acounter-example of L2, you musthave
  T(k v) ¹ k T(v) Shorthand notation for vectors.   Usethe following shorthand to enter your vectors:

  separate the entries by commas and enclose inc( ), e.g. c(1, 2 ,3) representsthe column vector
  é
  ê
  ê
  ê
  ë 1 2 3 ù
  ?
  ?
  ?
  û

T ?
ç
ç
ç
è é
ê
ê
ê
ë x y z ù
?
?
?
û ?
÷
÷
÷
? = é
ê
ê
ë 6 x-y+5 z -6 x-y+z+2 ù
?
?
û

Explanation / Answer

T(x,y,z) = ( 6x-y+5z , -6x-y+z+2) is not a lineartransformation and see that the constant other than 0 in thecomponents of vectors cannot be a linear transformation.

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