Question

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(AB^{T} ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2

3 4)

on U, and compute the minimal distance between A and an element of U.

Hint: Use the basis 1 0 0 0

0 0 0 1

0 1 1 0 of U.

Answer #1

A2. Let v be a fixed vector in an inner product space V. Let W
be the subset of V consisting of all vectors in V that are
orthogonal to v. In set language, W = { w LaTeX: \in
∈V: <w, v> = 0}. Show that W is a subspace of V. Then,
if V = R3, v = (1, 1, 1), and the inner product is the usual dot
product, find a basis for W.

Let V be a finite dimensional vector space over R with an inner
product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
(c) (2 points) Recall that the norm of a vector x ∈ V...

Let W be a subspace of a f.d. inner product space V and let PW
be the orthogonal projection of V onto W. Show that the
characteristic polynomial of PW is
(t-1)^dimW t^(dimv-dimw)

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

9. Let S and T be two subspaces of some vector space V.
(b) Define S + T to be the
subset of V whose elements have the form (an
element of S) + (an element of
T). Prove that S +
T is a subspace of V.
(c) Suppose {v1, . . . ,
vi} is a basis for the
intersection S ∩ T. Extend this with
{s1, . . . ,
sj} to a basis for
S, and...

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose that T is a linear transformation from V to itself and
T(u) = u + v,
T(v) = u, T(w) =
v.
1. Find the matrix of T relative to the ordered basis B.
2. A typical element of V looks like
au + bv +
cw, where a, b and c
are scalars. Find T(au +
bv + cw). Now
that you know...

3. a. Consider R^2 with the Euclidean inner product (i.e. dot
product). Let
v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v.
b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with
the Euclidean
inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7,
5).
C.Let V be an inner product space. Suppose u is orthogonal to
both v
and w. Prove that for any scalars c and d,...

We can identify the set V of all 3×3-matrices (real
coefficients) with vector space R9. Show that the set of all 3 × 3
symmetric matrices is a vector subspace of V .

Let Mn be the vector space of all n × n matrices with real
entries. Let W = {A ∈ M3 : trace(A) = 0}, U = {B ∈ Mn : B = B t }
Verify U, W are subspaces . Find a basis for W and U and compute
dim(W) and dim(U).

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