Question

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?

Answer #1

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,v5}
where v1= (1,−1,2,4), v2 = (0,3,1,2),
v3 = (3,0,7,14), v4 = (1,−1,2,0),
v5 = (2,1,5,6). Find a subset of S that forms a basis
for span(S).

. Let v1,v2,v3,v4 be a basis of V. Show that v1+v2, v2+v3,
v3+v4, v4 is a basis of V

If S is the set of vectors in R^4 (S= {v1, v2, v3, v4, v5})
where, v1 = (1,2,-1,1), v2 = (-3,0,-4,3), v3 = (2,1,1,-1), v4 =
(-3,3,-9,-6), v5 = (3,9,7,-6)
Find a subset of S that is a basis for the span(S).

If S=(v1,v2,v3,v4)
is a linearly independent sequence of vectors in Rn
then
A) n = 4
B) The matrix ( v1 v2 v3
v4) has a unique pivot column.
C) S is a basis for
Span(v1,v2,v3,v4)

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.

Determine whether the given vectors span R3
V1=(-1,5,2), V2=(3,1,1), V3=(2,0,-2), V4=(4,1,0)

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}

Let S = { e1, e2, e3,
e4 } be the standard basis for R4 , and let B
= { v1, v2, v3, v4, }
be the basis with vi = T(ei ), where
T ( x1, x2, x3, x4 )
= (x2, x3, x4, x1 ).
Find the transition matrices P B to S and P S to
B.

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...

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