Question

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.

Answer #1

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.

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}

let v1=[1,0,10], v2=[0,1,0,1] and let W be the
subspace of R^4 spanned by v1 and v2.
A. convert {v1,v2} into an orhonormal basis of W.
Basis =
B.find the projection of b=[-1,-2,-2,-1] onto W
C.find two linear independent vectors in R^4
perpendicular to W.
vectors =

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.

Let P2 denote the vector space of polynomials in x with real
coefficients having degree at most 2. Let W be a subspace of P2
given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper
subspace of P2.

Let S ∈ L(R2) be given by S(x1, x2) = (x1 +x2, x2) and let I ∈
L(R2) be the identity operator.
Using the inner product defined in problem 1 for the standard basis
and the dot product,
compute <S, I>, || S ||, and || I ||
{Inner product in problem 1: Let W be an inner product space and
v1, . . . , vn a basis of V. Show that <S, T> = <Sv1, T
v1> +...

Let the set W be: all polynomials in P3 satisfying
that p(-t)=p(t),
Question: Is W a vector space or not?
If yes, find a basis and dimension

Consider the subspace of ? = ?0(?) spanned by {?1(?) = 1 − 2x,
?2(?) = x2 } over the interval [0, 1]
a. Determine if ?1 is orthogonal to ?2. (10pts)
b. Orthogonalize {? 1, ? 2} using the Gram-Schmidt process.
(10pts)

Roll two dice (one red and one white). Denote their outcomes as
X1 and X2. Let T = X1+X2 denote the total, let X1 W X2 denote the
maximum and let X1 V X2 denote the minimum. Find the following
probabilities: (a) P(X1 ≥ 3|X2 ≤ 4) (b) P(T is prime) (c) P(T ≤
8|X1 W X2 = 5) (d) P(X1 V X2 ≤ 5|T ≥ 8) (e) P(X1 W X2 ≥ 3|X1 W X2 ≤
3)

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

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