Page 384 - 35Linear Algebra

384 Movie Scripts

 
a 0
2
a 0 + a 1 t + a 2 t as  a 1 
a 2
And think for a second about how you add polynomials, you match up terms of
the same degree and add the constants component-wise. So it makes some sense
to think about polynomials this way, since vector addition is also component-
wise.
We could also write the output
 
b 0
3
2
b 0 + b 1 t + b 2 t + b 3 t as  b 1  b 3
b 2
Then lets look at the information given in the problem and think about it
in terms of column vectors
2
• L(1) = 4 but we can think of the input 1 = 1 + 0t + 0t and the output
 
  4
1
0
2 3  

4 = 4 + 0t + 0t 0t and write this as L( 0  ) =  
 0 
0
0
 
  0
0
0
3  
• L(t) = t This can be written as L( 1  ) =  

 0 
0
1
2
• L(t ) = t − 1 It might be a little trickier to figure out how to write
t − 1 but if we write the polynomial out with the terms in order and with
zeroes next to the terms that do not appear, we can see that
 
−1
1 

2 3
t − 1 = −1 + t + 0t + 0t corresponds to  
 0 
0
 
−1
 
0
1
 
) =  
So this can be written as L( 0 

 0 
1
0
Now to think about how you would write the linear transformation L as
a matrix, first think about what the dimensions of the matrix would be.
Then look at the first two parts of this problem to help you figure out
what the entries should be.
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