In: Advanced Math

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal basis {v1,v2.....vk-1}. Work out a orthonormal basis of W in terms of that of V and the orthogonal complement of V in W.

Provide detailed reasoning.

Let V -Φ -> W be linear. Show that ker (Φ) is a
subspace of V and Φ (V) is a subspace of W.

Find a basis and the dimension of the subspace:
V = {(x1, x2, x3,
x4)| 2x1 = x2 + x3,
x2 − 2x4 = 0}

Determine whether or not W is a subspace of V. Justify your
answer.
W = {p(x) ∈ P(R)|p(1) = −p(−1)}, V =
P(R)

Let V be a vector space of dimension 1 over a field k and choose
a fixed nonzero element voe V, which is therefore a basis. Let W be
any vector space over k and let woe W be an arbitrary vector. Show
that there is a unique linear transformation T: V → W such that
T(v)= wo. [Hint: What must T(Avo) be?)

Let W be a subspace of Rn. Prove that W⊥ is also a
subspace of Rn.

consider the subspace
W=span[(4,-2,1)^T,(2,0,3)^T,(2,-4,-7)^T]
Find
A) basis of W
B) Dimension of W
C) is vector v=[0,-2,-5]^T contained in W? if yes espress as
linear combantion

Prove the following:
Let V and W be vector spaces of equal (finite) dimension, and
let T: V → W be linear. Then the following are equivalent.
(a) T is one-to-one.
(b) T is onto.
(c) Rank(T) = dim(V).

1. For a map f : V ?? W between vector spaces V and W to be a
linear map it must preserve the structure of V . What must one
verify to verify whether or not a map is linear?
2. For a map f : V ?? W between vector spaces to be an
isomorphism it must be a linear map and also have two further
properties. What are those two properties? As well as giving the
names...

Let W be a discrete random variable and Pr(W = k) = 1/6, k = 1,
2 ,....., 6. Define
X =
{
W, if W <= 3;
1, if W >= 4;
}
and Y =
{
3, if W <= 3;
7 -W, if W >= 4;
}
(a) Find the joint probability mass function of (X, Y ) and compute
Pr(X +Y = 4).
(b) Find the correlation Cor(X, Y ). Are X and Y independent?
Explain.

Determine whether or not W is a subspace of R3 where
W consists of all vectors (a,b,c) in R3 such that
1. a =3b
2. a <= b <= c
3. ab=0
4. a+b+c=0
5. b=a2
6. a=2b=3c

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