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**Unformatted text preview: **{1 , . . . , k } be a linearly independent set in Rn . Prove that if ∈ span B ,
v
v
v
then {1 , . . . , k , } is linearly independent.
v
vv
11. Prove or disprove each of the following statements
a) The rank of a matrix is the number of rows in the RREF of the matrix that are
composed entirely of zeroes. 1
1 is a solution of a homogeneous system of linear equations, then the system has
b) If
1
inﬁnitely many solutions.
c) Let , ∈ Rn , = 0. If · = · , then = .
a b, c
a
ab ac
bc
d) Let , ∈ R3 such that { , } is linearly independent. If w is orthogonal to and ,
uv
uv
u
v
then { , , w} is also linearly independent.
uv
e) Let ∈ R3 . If = then the set with vector equation = + t is not a subspace
b, v
b 0,
xb
v
of R3 .
2...

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