Suppose that ? and ? are subspaces of a vector space ? with ? =
?...
Suppose that ? and ? are subspaces of a vector space ? with ? =
? ⊕ ?. Suppose also that ??, … , ?? is a basis of ? amd ??, … , ??
is a basis of ?. Prove ??, … , ??, ??, … , ?? is a basis of V.
Which of the following are subspaces of the vector space of
real-valued functions of a real variables? (must select all of the
subspaces.)
A. The set of even function (f(-x) = f(x) for all numbers
x).
B. The set of odd functions (f(-x) = -f(x) for all real numbers
x).
C. The set of functions f such that f(0) = 7
D. The set of functions f such that f(7) = 0
(1) Suppose that V is a vector space and that S = {u,v} is a set
of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so
thatT is another set of two vectors in V ). (a) Show that if S is
linearly independent in V then T is also independent. (Hint:
suppose that there is a linear combination of elements of T that is
equal to 0. Then ....). (b) Show that if S generates V...
suppose that T : V → V is a linear map on a finite-dimensional
vector space V such that dim range T = dim range T2. Show that V =
range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range
T = {0}, and apply the fundamental theorem of linear maps.)
(3) Let V be a vector space over a field F. Suppose that a ? F,
v ? V and av = 0. Prove that a = 0 or v = 0.
(4) Prove that for any field F, F is a vector space over F.
(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1,
a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...
Let T be an operator on a finite-dimensional complex vector
space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that
T does not have a square root; i.e. there does not exist any S ∈ L
(V) such that S2 = T.