2
Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and
w=(3,5,4,4).
2.1
Construct a basis for the vector space spanned by u, v and w.
2.2
Show that c=(1,3,2,1) is not in the vector space spanned by the
above vectors u,v and w.
2.3
Show that d=(4,9,17,-11) is in the vector space spanned by the
above vectors u,v and w, by expressing d as a linear combination of
u,v and w.
Let u and v be vectors in
R3. Consider the following statements.T or
F
(1) |u · v|
≤ ||u|| + ||v||
(2) If au +
bv = cu +
dv then a = c and
b = d.
(3) ||u + v||2 =
||u||2 +
||v||2 +
2(u · v)
Let u, v, and
w be vectors in R3. T or
F.
(1) u · v −
||u||
(2)
(u · v) × w
(3)
|| ( ||u|| projvu ...
Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4,
∥u + v∥ = 5. Find the inner product 〈u, v〉.
Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that
{a1, · · · ak} is a linearly independent set.
The Cauchy-Schwarz Inequality Let u and v be vectors in R 2
.
We wish to prove that -> (u · v)^ 2 ≤ |u|^
2 |v|^2 .
This inequality is called the Cauchy-Schwarz inequality and is
one of the most important inequalities in linear algebra.
One way to do this to use the angle relation of the dot product
(do it!). Another way is a bit longer, but can be considered an
application of optimization. First, assume that the...
Let ?V be the set of vectors in ?2R2 with the following
definition of addition and scalar multiplication:
Addition: [?1?2]⊕[?1?2]=[0?2+?2][x1x2]⊕[y1y2]=[0x2+y2]
Scalar Multiplication: ?⊙[?1?2]=[0??2]α⊙[x1x2]=[0αx2]
Determine which of the Vector Space Axioms are satisfied.
A1. ?⊕?=?⊕?x⊕y=y⊕x for any ?x and ?y in ?V
? YES NO
A2. (?⊕?)⊕?=?⊕(?⊕?)(x⊕y)⊕z=x⊕(y⊕z) for any ?,?x,y and ?z in
?V
? YES NO
A3. There exists an element 00 in ?V such that ?⊕0=?x⊕0=x for
each ?∈?x∈V
? YES NO
A4. For each ?∈?x∈V, there exists an...
show that for any two vectors u and v in an inner product
space
||u+v||^2+||u-v||^2=2(||u||^2+||v||^2)
give a geometric interpretation of this result fot he vector
space R^2
Let U and V be vector spaces, and let L(V,U) be the set of all
linear transformations from V to U. Let T_1 and T_2 be in
L(V,U),v be in V, and x a real number. Define
vector addition in L(V,U) by
(T_1+T_2)(v)=T_1(v)+T_2(v)
, and define scalar multiplication of linear maps as
(xT)(v)=xT(v). Show that under
these operations, L(V,U) is a vector space.