Let H=Span{v1,v2} and
K=Span{v3,v4}, where
v1,v2,v3,v4 are given
below.
v1 = [3 2 5], v2 =[4...
Let H=Span{v1,v2} and
K=Span{v3,v4}, where
v1,v2,v3,v4 are given
below.
v1 = [3 2 5], v2 =[4 2 6], v3
=[5 -1 1], v4 =[0 -21 -9]
Then H and K are subspaces of R3 . In fact, H and K
are planes in R3 through the origin, and they intersect
in a line through 0. Find a nonzero vector w that
generates that line.
w = { _______ }
Let S = {v1, v2, v3, v4} be a given basis of R ^4 . Suppose...
Let S = {v1, v2, v3, v4} be a given basis of R ^4 . Suppose that
A is a (3 × 4) matrix with the following properties: Av1 = 0, A(v1
+ 2v4) = 0, Av2 =[ 1 1 1 ] T , Av3 = [ 0 −1 −4
]T . Find a basis for N (A), and a basis for R(A). Fully
justify your answer.
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.
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in...
Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in the span of S? Why?
let v1=[1,0,10], v2=[0,1,0,1] and let W be the
subspace of R^4 spanned by v1 and v2....
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 =
Determine if the vectors v1= (3, 0, -3, 6),
v2 = ( 0, 2, 3, 1),...
Determine if the vectors v1= (3, 0, -3, 6),
v2 = ( 0, 2, 3, 1), and v3 = (0, -2, 2, 0 )
form a linearly dependent set in R 4. Is it a basis of
R4 ?
Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set,...
Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a
linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W =
span{w⃗1,w⃗2,w⃗3}.
(a) Is there a linear transformation P : V → W such that P(⃗vi)
= w⃗i for i = 1, 2, 3?
(b) Is there a linear transformation Q : W → V such that Q(w⃗i)
= ⃗vi for i = 1, 2, 3?
Hint: the...
Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 =
[a,1,0,b], and v4 = [3,2,a+b,0],...
Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 =
[a,1,0,b], and v4 = [3,2,a+b,0], where a and b are parameters. Find
all conditions on the values of a and b (if any) for which:
1. The number of linearly independent vectors in this collection
is 1.
2. The number of linearly independent vectors in this collection
is 2.
3. The number of linearly independent vectors in this collection
is 3.
4. The number of linearly independent vectors in...
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?