Question

Find the dimension of the subspace of R5 consisting of all vectors of the form (a, b, c, d, e) where a = 2b and c = 4d.

Answer #1

Let W be the subset of R^R consisting of all functions of the
form x ?→a · cos(x − b), for real numbers a and b. Show that W is a
subspace of R^R and find its dimension.

Let V be the subspace of all vectors in R 5 , such that x1 − x4
= x2 − 5x5 = 3x3 + x4
(a) Find a matrix A with that space as its Null space; What is
the rank of A?
b) Find a basis B1 of V ; What is the dimension of V ?
(c) Find a matrix D with V as its column space. What is the rank
of D? To find the rank of...

Use Theorem on Subspaces to determine whether the set of all
vectors of the form (b, 1, a) form a subspace of R3.

find the basis and dimension for the span of each of
the following sets of vectors.
a={[2,-1,1],[0,0,0],[-4,2,-2],[6,-3,3]}
basis=
dimension=
b={[3,3,3],[9,9,10],[21,21,23],[-33,-33,-36]}
basis=
dimension=

Let G be the subgroup of R^3 consisting of all vectors of the
form (x, y, 0). Let G act on R^3 by left multiplication. Describe
the orbits of this G-action geometrically. Show that the set of
orbits are in one to one correspondence with R

Find a linearly independent set of vectors that spans the same
subspace of R3 as that spanned by the vectors
[-3,1,3] , [-6,5,5],[0,-3,1]
Linearly independent set:
[x,y,z] , [x,y,z]

Find the dimension of the subspace U = span {1,sin^2(θ), cos 2θ}
of F[0, 2π]

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

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

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