Page 328 - 35Linear Algebra

328 Sample First Midterm

I kI
k
so, M = , or explicitly
0 I
 
1 0 0 k 0 0
0 1 0 0 k 0
 
 
0 0 1 0 0 k 
k

M =   .
 0 0 0 1 0 0 
 
 0 0 0 0 1 0 
0 0 0 0 0 1
7. We can call a function f : V −→ W linear if the sets V and W are vector
spaces and f obeys
f(αu + βv) = αf(u) + βf(v) ,

for all u, v ∈ V and α, β ∈ R.
Now, integration is a linear transformation from the space V of all inte-
grable functions (don’t be confused between the definition of a linear func-
tion above, and integrable functions f(x) which here are the vectors in V )
R ∞ R ∞
to the real numbers R, because (αf(x) + βg(x))dx = α f(x)dx +
R ∞ −∞ −∞
β g(x)dx.
−∞
8. The four main ingredients are (i) a set V of vectors, (ii) a number field K
(usually K = R), (iii) a rule for adding vectors (vector addition) and (iv)
a way to multiply vectors by a number to produce a new vector (scalar
multiplication). There are, of course, ten rules that these four ingredients
must obey.

(a) This is not a vector space. Notice that distributivity of scalar multi-
plication requires 2u = (1 + 1)u = u + u for any vector u but

a b 2a b
2 · =
c d 2c d
which does not equal

a b a b 2a 2b
+ = .
c d c d 2c 2d
This could be repaired by taking

a b ka kb
k · = .
c d kc kd
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