Question

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).

Answer #1

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).

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 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 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 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.

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 = (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?

Find a subset of the given vectors that form a basis for the
space spanned by the vectors. Verify that the vectors you chose
form a basis by showing linear independence and span: v1
(1,3,-2), v2 (2,1,4), v3(3,-6,18),
v4(0,1,-1), v5(-2,1-,-6)

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}

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