convert the basis V1=(1,-1,0), v2=(0,1,-1), v3=(-1,1,-1)for R^3
into an orthonormal basis, using theGram-Schmidt process and the...
convert the basis V1=(1,-1,0), v2=(0,1,-1), v3=(-1,1,-1)for R^3
into an orthonormal basis, using theGram-Schmidt process and the
standard inner product in R^3
5. Let v1 = (1/3,−2/3,2/3), v2 = (2/3,−1/3,−2/3) and v3 =
(2/3,2/3,1/3).
(a) Verify that v1,...
5. Let v1 = (1/3,−2/3,2/3), v2 = (2/3,−1/3,−2/3) and v3 =
(2/3,2/3,1/3).
(a) Verify that v1, v2, v3 is an orthonormal basis of R 3 .
(b) Determine the coordinates of x = (9, 10, 11), v1 − 4v2 and
v3 with respect to v1, v2, v3.
Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector...
Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun
where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>
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}
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in...
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in the span of S? Why?
Let
{V1, V2,...,Vn} be a linearly independent set of vectors choosen
from vector space V. Define...
Let
{V1, V2,...,Vn} be a linearly independent set of vectors choosen
from vector space V. Define w1=V1, w2= v1+v2, w3=v1+ v2+v3,...,
wn=v1+v2+v3+...+vn.
(a) Show that {w1, w2, w3...,wn} is a linearly independent
set.
(b) Can you include that {w1,w2,...,wn} is a basis for V? Why
or why not?
Let V be a finite dimensional vector space over R with an inner
product 〈x, y〉...
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 V be the vector space of 2 × 2 matrices over R, let <A,
B>=...
Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) 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...
Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be...
Let V be a vector space: d) Suppose that V is
finite-dimensional, and let S be a set of inner products on V that
is (when viewed as a subset of B(V)) linearly independent. Prove
that S must be finite
e) Exhibit an infinite linearly independent set of inner
products on R(x), the vector space of all polynomials with real
coefficients.
Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4...
Let
R4
have the inner product
<u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4
(a)
Let w = (0, 6,
4, 1). Find ||w||.
(b)
Let W be the
subspace spanned by the vectors
u1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).
Use the Gram-Schmidt process to transform the basis
{u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas.