If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and...
If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...
(i) Let u= (u1,u2) and v=
(v1,v2). Show that the following is an inner
product by...
(i) Let u= (u1,u2) and v=
(v1,v2). Show that the following is an inner
product by verifying that the inner product hold
<u,v>= 4u1v1 +
u2v2 +4u2v2
(ii) Let u= (u1,
u2, u3) and v=
(v1,v2,v3). Show that the
following is an inner product by verifying that the inner product
hold
<u,v> =
2u1v1 + u2v2 +
4u3v3
Suppose u = (u1,u2) and v = (v1, v2) are two vectors in R2.
Explain why...
Suppose u = (u1,u2) and v = (v1, v2) are two vectors in R2.
Explain why the operations (u * v) = u1v2 cannot be an inner
product.
5. Let U1, U2, U3 be subspaces
of a vector space V.
Prove that U1, U2,...
5. Let U1, U2, U3 be subspaces
of a vector space V.
Prove that U1, U2, U3 are
direct-summable if and only if
(i) the intersection of U1 and U2 is 0.\,
and
(ii) the intersection of U1+U2 and
U3 is 0.
A detailed explanation would be greatly appreciated :)
Determine if the vector v is a linear combination of the vectors
u1, u2, u3. If...
Determine if the vector v is a linear combination of the vectors
u1, u2, u3. If yes, indicate at least one possible value for the
weights. If not, explain why.
v =
2
4
2
, u1 =
1
1
0
, u2 =
0
1
-1
, u3 =
1
2
-1
Use Gram-Schmidt process to transform the basis {u1, u2, u3},
where u1=(1,1,1), u2=(1,2,0), u3=(1,0,-1),:
a) for...
Use Gram-Schmidt process to transform the basis {u1, u2, u3},
where u1=(1,1,1), u2=(1,2,0), u3=(1,0,-1),:
a) for the Euclidean IPS. (IPS means inner product space)
Let B1 = { u1, u2, u3 }, where u1 = (2,?1, 1), u2 = (1,?2,...
Let B1 = { u1, u2, u3 }, where u1 = (2,?1, 1), u2 = (1,?2, 1),
and u3 = (1,?1, 0). B1 is a basis for R^3 .
A. Find the transition matrix Q ^?1 from the standard basis of R
^3 to B1 .
B. Write U as a linear combination of the basis B1 .
Linear Algebra
Write x as the sum of two vectors, one is Span {u1,
u2, u3}...
Linear Algebra
Write x as the sum of two vectors, one is Span {u1,
u2, u3} and one in Span {u4}. Assume that
{u1,...,u4} is an orthogonal basis for
R4
u1 = [0, 1, -6, -1] , u2 = [5, 7, 1, 1],
u3 = [1, 0, 1, -6], u4 = [7, -5, -1, 1], x =
[14, -9, 4, 0]
x =
(Type an integer or simplified fraction for each matrix
element.)
Prove that
Let S={v1,v2,v3} be a linearly indepedent set of vectors om a
vector space V....
Prove that
Let S={v1,v2,v3} be a linearly indepedent set of vectors om a
vector space V. Then so are
{v1},{v2},{v3},{v1,v2},{v1,v3},{v2,v3}
Suppose the vectors v1, v2, . . . , vp span a vector space V
....
Suppose the vectors v1, v2, . . . , vp span a vector space V
.
(1) Show that for each i = 1, . . . , p, vi belongs to V ;
(2) Show that given any vector u ∈ V , v1, v2, . . . , vp, u also
span V