Page 300 - 35Linear Algebra

300 Kernel, Range, Nullity, Rank

Hint

3. Let {v 1 , . . . , v n } be a basis for V and L : V → W is a linear function.
Carefully explain why

L(V ) = span{Lv 1 , . . . , Lv n } .

4
3
4. Suppose L: R → R whose matrix M in the standard basis is row
equivalent to the following matrix:
 
1 0 0 −1
0 1 0 1 = RREF(M) ∼ M.
 
0 0 1 1

(a) Explain why the first three columns of the original matrix M form
4
a basis for L(R ).
(b) Find and describe an algorithm (i.e., a general procedure) for
m
n
n
computing a basis for L(R ) when L: R → R .
4
4
3
(c) Use your algorithm to find a basis for L(R ) when L: R → R is
the linear transformation whose matrix M in the standard basis
is
 
2 1 1 4
0 1 0 5 .
 
4 1 1 6
5. Claim:
If {v 1 , . . . , v n } is a basis for ker L, where L: V → W, then it
is always possible to extend this set to a basis for V .

Choose some simple yet non-trivial linear transformations with non-
trivial kernels and verify the above claim for those transformations.

6. Let P n (x) be the space of polynomials in x of degree less than or equal
to n, and consider the derivative operator

d
: P n (x) → P n (x) .
dx

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