Page 323 - 35Linear Algebra
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8. What are the four main things we need to define for a vector space? Which
of the following is a vector space over R? For those that are not vector
spaces, modify one part of the definition to make it into a vector space.
(a) V = { 2 × 2 matrices with entries in R}, usual matrix addition, and
a b ka b
k · = for k ∈ R.
c d kc d
(b) V = {polynomials with complex coefficients of degree ≤ 3}, with usual
addition and scalar multiplication of polynomials.
3
(c) V = {vectors in R with at least one entry containing a 1}, with usual
addition and scalar multiplication.
9. Subspaces: If V is a vector space, we say that U is a subspace of V when the
set U is also a vector space, using the vector addition and scalar multiplica-
tion rules of the vector space V . (Remember that U ⊂ V says that “U is a
subset of V ”, i.e., all elements of U are also elements of V . The symbol ∀
means “for all” and ∈ means “is an element of”.)
Explain why additive closure (u + w ∈ U ∀ u, v ∈ U) and multiplicative
closure (r.u ∈ U ∀ r ∈ R, u ∈ V ) ensure that (i) the zero vector 0 ∈ U and
(ii) every u ∈ U has an additive inverse.
In fact it suffices to check closure under addition and scalar multiplication
to verify that U is a vector space. Check whether the following choices of U
are vector spaces:
x
y
(a) U = : x, y ∈ R
0
1
(b) U = : z ∈ R
0
z
10. Find an LU decomposition for the matrix
1 1 −1 2
1 3 2 2
−1 −3 −4 6
0 4 7 −2
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