Question

Consider the subspace of ? = ?0(?) spanned by {?1(?) = 1 − 2x,
?2(?) = x^{2} } over the interval [0, 1]

a. Determine if ?1 is orthogonal to ?2. (10pts)

b. Orthogonalize {? 1, ? 2} using the Gram-Schmidt process. (10pts)

Answer #1

Let W be the subspace of R4
spanned by the vectors
u1 = (−1, 0, 1, 0),
u2 = (0, 1, 1, 0), and
u3 = (0, 0, 1, 1).
Use the Gram-Schmidt process to transform the basis
{u1, u2,
u3} into an orthonormal basis.

Use the Gram-Schmidt process to find an orthonormal basis for
the subspace of R4 spanned by the vectors
u1 = (1, 0, 0, 0), u2 = (1, 1, 0, 0),
u3 = (0, 1, 1, 1).
Show all your work.

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

Consider interval [0,2] and an inner product with weight w(x)=1.
Starting with 1, x, x^2, x^3 and using Gram-Schmidt process, build
four polynomials of degree 0,1,2,3 orthogonal on [0,2]. You do not
have to normalize them (I think this simplifies calculations). Just
remember that if they are not normalized, Gram-Schmidt formula will
have denominators in the form <q_j,q_j>. (If you do normalize
them, then <q_j,q_j> = 1).

Let W⊂ C1 be the subspace spanned by the two polynomials x1(t) =
1 and x2(t) =t^2. For the given function y(t)=1−t^2 decide whether
or not y(t) is an element of W. Furthermore, if y(t)∈W, determine
whether the set {y(t), x2(t)} is a spanning set for W.

Find the orthogonal projection of v⃗ =⎢4,−11,−36,9⎤ onto the
subspace W spanned by ⎢0,0,−5,−2| , |−4,2,5,−5⎢ , ⎢−5,−5,0,5|

Find the orthogonal projection of v=[−2,10,−16,−19] onto the
subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

a) Apply the Gram–Schmidt process to find an orthogonal basis
for S.
S=span{[110−1],[1301],[4220]}
b) Find projSu.
S = subspace in Exercise 14; u=[1010]
c) Find an orthonormal basis for S.
S= subspace in Exercise 14.

Let W be a subspace of R^4 spanned by v1 =
[1,1,2,0] and v2 = 2,-1,0,4]. Find a basis for W^T
= {v is in R^2 : w*v = 0 for
w inside of W}

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