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?
A. Suppose that v1, v2, v3 are linearly independant and
w1=v1+v2, w2=v2-v3, w3= v2+v3. Determine whether...
A. Suppose that v1, v2, v3 are linearly independant and
w1=v1+v2, w2=v2-v3, w3= v2+v3. Determine whether w1, w2, w3 are
linear independent or linear deppendent.
B. Find the largest possible number of independent vectors
among:
v1=(1,-1,0,0), v2=(1,0,-1,0), v3=(1,0,0,-1), v4=(0,1,-1,0),
v5=(0,1,0,-1), v6=(0,0,1,-1)
Let W be the subspace of R4
spanned by the vectors
u1 = (−1, 0, 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.
(a) Use the Gram-Schmidt process on the basis {(1, 2, 2),(1, 2,
3),(4, 3, 2)} of...
(a) Use the Gram-Schmidt process on the basis {(1, 2, 2),(1, 2,
3),(4, 3, 2)} of R ^3 find an orthonormal basis.
(b) Write the vector v = (2, 1, −5) as a linear combination of
the orthonormal basis vectors found in part (a).
Applythe Gram-schmidt orthonormalizaion proceudre to the set
1,x2,x4 which is linearly independent on the
interval [1,2]...
Applythe Gram-schmidt orthonormalizaion proceudre to the set
1,x2,x4 which is linearly independent on the
interval [1,2] to construct a mutually orthonormal system phi_1(x),
phi_2(x), phi_3(x)
Let S = {?2, (? − 1)2, (? − 2)2 }
A) Show that S forms...
Let S = {?2, (? − 1)2, (? − 2)2 }
A) Show that S forms a basis for P2
.
B) Define an inner product on P2 via < p(x) | q(x) > =
p(-1)q(-1) + p(0)q(0) + p(1)q(1).
Using this inner product, and Gram-Schmidt, construct an
orthonormal basis for P2 from S – use the
vectors in the order given!
If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and...
If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...