Question
Transcribed Text
1. Determine Whether the set with given operation defines a vector space. If not, indicate which law(s)
fail.
(a) V = with standard matrix addition and scalar multiplication
(b) V consists of all continuous functions f(x) on the interval [0, 1) such that f(0) = 0 with standard
function addition and scalar multiplication. (V = {J(x) € C(0,11/5(0) = 0}).
(c) V = {(x, fx)(r € IR} with the standard vector addition and scalar multiplication.
(d) Let V be the set C3 with the standard vector addition, but with scalar multiplication defined by
2
az
a
y
=
oz
z
ag
(e) V is the set of all fifth-degree polynomials with standard operations.
2. Determine whether the given subset W is a subspace of the vector space V. Justify your answer.
(a) V = R3 and W = {(a,0, a - b)|a, b € R}
(b) V = C10, 1] and W = {J(x) € C10, < 0).
(c) V = CS and W = 6 5
3. Find the redundant vectors, if any in the following sets.
(a) S = {r,5x}}
(b) S = (2,2-2,27,1+224
(c) S =
4. Assume V is a vector space of dimension 72 and S = (U1, V vit CV. Answer True or False
for
the
following statements.
(a) S is either a basis or contains redundant vectors.
(b) A linearly independent set contains no redundant vectors.
(c) If V = (zy. vg) and dim V = 2, then the set is a linearly dependent set.
(d) A set of vectors containing the zero vector is linearly independent set.
(e) Every vector space is finite-dimensional.
(f) The set of vectors { : i : 0 1 in C2 contains redundant
vectors.
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