Page 119 - 35Linear Algebra

6.5 Review Problems 119

2. If f is a linear function of one variable, then how many points on the
graph of the function are needed to specify the function? Give an
explicit expression for f in terms of these points. (You might want
to look up the definition of a graph before you make any assumptions
about the function.)

1 2
3. (a) If p = 1 and p = 3 is it possible that p is a linear
2 4
function?
2
4
2
3
(b) If Q(x ) = x and Q(2x ) = x is it possible that Q is a linear
function from polynomials to polynomials?
4. If f is a linear function such that

1 2
f = 0, and f = 1 ,
2 3

x
then what is f ?
y
5. Let P n be the space of polynomials of degree n or less in the variable t.
Suppose L is a linear transformation from P 2 → P 3 such that L(1) = 4,
2
3
L(t) = t , and L(t ) = t − 1.
2
(a) Find L(1 + t + 2t ).
2
(b) Find L(a + bt + ct ).
2
3
(c) Find all values a, b, c such that L(a + bt + ct ) = 1 + 3t + 2t .

Hint

6. Show that the operator I that maps f to the function If defined
R x
by If(x) := f(t)dt is a linear operator on the space of continuous
0
functions.
7. Let z ∈ C. Recall that z = x+iy for some x, y ∈ R, and we can form the
2 2
complex conjugate of z by taking z = x − iy. The function c: R → R
which sends (x, y) 7→ (x, −y) agrees with complex conjugation.

(a) Show that c is a linear map over R (i.e. scalars in R).
(b) Show that z is not linear over C.

119