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LAHW#10
Due November 29, 2010
5.1 Column, Row, and Null Spaces
• 6.
– Determine whether the set of all 2×2 matrices
of the form
b
 a
b  a c 


is a subspace R2×2.
5.1 Column, Row, and Null Spaces
• 15.
– Is the union of two subspace in a vector
space always a subspace? Explain why or
find a counterexample. The union is defined
by U∪W = {x: x∈U or x∈W}. (In mathematics,
we always use the inclusive or. Thus, p or q is
true when either p is true or q is true or both
are true.)
5.1 Column, Row, and Null Spaces
• 16. (Continuation.)
– Find the exact conditions on subspaces U and
W in a vector space V in order that their union
be a subspace.
5.1 Column, Row, and Null Spaces
• 18.
– Fix a set of vectors {u1, u2, …, un} in some
vector space. Explain why
the set of n-tuples
n
(c1, c2, …, cn) such that  ciui  0 is a subspace
i 1
n
of R .
5.1 Column, Row, and Null Spaces
• 24.
– Establish that the span of a set in a vector
space is the smallest subspace containing
that set.
5.1 Column, Row, and Null Spaces
• 32.
– If f : X  Y and if U  X , then we can create a new function g
by the equation g ( x)  f ( x) for all x in U . If U is a proper subset
of X , then g  f because they have different domains. The function
g is called the restrictio n of f to U . The nonation g  f | U is often
used. Suppose now that X and Y are linear spaces and L is a linear
map from X to Y . Let U be a subapsce of X . Is L | U linear? What
is its domain? Do L and L | U have the same range?
(Arguments and examples are needed.)
5.1 Column, Row, and Null Spaces
• 39.
– Use A =
 5 0 0 10
  1 0 1 4


in this exercise.
a. Find a set of vectors that spans the column space.
b. Is [-1, 1]T in the column space?
c. Find a set of vectors that spans the row space.
d. Is [-1, 1, 1, 4] in the row space?
e. Find a set of vectors that span the null space.
f. Is [-2, 1, -6, 1]T in the null space?
5.1 Column, Row, and Null Spaces
• 40.
– Is the set of all vector x = [x1, x2, x3, x4]T that
are linear combinations of vectors [4, 2, 0, 1]T
and [6, 3, -1, 2]T, and in addition satisfy the
equation x1 = 2x2 a subspace of R4? Explain
why or why not.