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 æ
ç
ç
ç
ç
ç
è é
ê
ê
ê
ê
ê
ë x1 x2 x3 x4 ù
ú
ú
ú
ú
ú
û ö
÷
÷
÷
÷
÷
ø = é
ê
ê
ê
ë 6 x1-x2+x3 5 x1-2 x2-4 x3-3 x4 3 x1-4 x2+6 x3-3 x4+6 ù
ú
ú
ú
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
  ú
  ú
  ú
  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
  ú
  ú
  ú
  û

T æ
ç
ç
ç
ç
ç
è é
ê
ê
ê
ê
ê
ë x1 x2 x3 x4 ù
ú
ú
ú
ú
ú
û ö
÷
÷
÷
÷
÷
ø = é
ê
ê
ê
ë 6 x1-x2+x3 5 x1-2 x2-4 x3-3 x4 3 x1-4 x2+6 x3-3 x4+6 ù
ú
ú
ú
û

Explanation / 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
  ú
  ú
  ú
  û
  

T æ
ç
ç
ç
ç
ç
è é
ê
ê
ê
ê
ê
ë x1 x2 x3 x4 ù
ú
ú
ú
ú
ú
û ö
÷
÷
÷
÷
÷
ø = é
ê
ê
ê
ë 6 x1-x2+x3 5 x1-2 x2-4 x3-3 x4 3 x1-4 x2+6 x3-3 x4+6 ù
ú
ú
ú
û

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