suppose {V1,
V2 , V3 }
is a pairwise orthogonal set of nonzero vectors in Rn....
suppose {V1,
V2 , V3 }
is a pairwise orthogonal set of nonzero vectors in Rn.
Show that {V1,
V2 , V3 }
is also linear independent.
Determine whether the given vectors span R3
V1=(-1,5,2), V2=(3,1,1), V3=(2,0,-2), V4=(4,1,0)
Determine whether the given vectors span R3
V1=(-1,5,2), V2=(3,1,1), V3=(2,0,-2), V4=(4,1,0)
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of
vectors in V...
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of
vectors in V , and if ⃗v4 ∈ V , then {⃗v1, ⃗v2, ⃗v3, ⃗v4} is
also a linear dependent set of vectors in V .
2. Prove that if {⃗v1,⃗v2,...,⃗vr} is a linear dependent set of
vectors in V, and if⃗ vr + 1 ,⃗vr+2,...,⃗vn ∈V, then
{⃗v1,⃗v2,...,⃗vn} is also a linear dependent set of vectors in
V.
Determine all real numbers a for which the vectors
v1 = (1,−1,1,a,2)
v2 = (−1,0,0,1,0)
v3...
Determine all real numbers a for which the vectors
v1 = (1,−1,1,a,2)
v2 = (−1,0,0,1,0)
v3 = (1,2,a + 1,1,0)
v4 = (2,0,a + 3,2a + 3,4)
make a linearly independent set. For which values of a does the
set contain at least three linearly independent vectors?
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.
Exercise 6. Consider the following vectors in R3 . v1 = (1, −1,
0) v2 =...
Exercise 6. Consider the following vectors in R3 . v1 = (1, −1,
0) v2 = (3, 2, −1) v3 = (3, 5, −2 ) (a) Verify
that the general vector u = (x, y, z) can be written as a linear
combination of v1, v2, and v3. (Hint : The coefficients will be
expressed as functions of the entries x, y and z of u.) Note : This
shows that Span{v1, v2, v3} = R3 . (b) Can R3 be...
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)
Do the vectors v1 = 1 2 3 ,
v2 = ...
Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...
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}
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3...
(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...