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Linear Independence, Basis, Dimension and Co-ordinates

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1. Consider the vector space

R

² over

R

. Then (2, 7) can be written as a linear combination of elements of which of the following sets?

a:

S

₁={(1, 3), (2, 6)} Explanation

Explanation

b:

S

₂={(1, 0), (0, 0), (2, 0))} Explanation

Explanation

c:

S

₃={(0, 1), (1, 2), (1, 1)} Explanation

Explanation

d:

S

₄={(1, 2), (0, 0), (2, 0))} Explanation

Explanation

2. Consider the vector space

P

₂(

x

) over

R

.Then which of the following sets can span

P

₂(

x

)?

a:

S

={1, 2

x

,

x

₂+2} Explanation

Explanation

b:

S

={

x

-

x

², -

x

,

x

²} Explanation

Explanation

c:

S

={1, 0,

x

²-4} Explanation

Explanation

d:

S

={1, -5

x

} Explanation

Explanation

3. Consider the vector space

R

³ over

R

.Then which of the following sets can span

R

³?

a:

S

={(2, 0, -2), (3, 1, 1), (2, 0, 0) (0, 0, 1)} Explanation

Explanation

b:

S

={(1, 0, 2), (1, -1, 1), (0, 0, 0)} Explanation

Explanation

c:

S

={(-1, 0, 1), (1, 0, 0) (0, 0, 1)} Explanation

Explanation

d:

S

={(1, 0, 0), (2, 1, 0) (0, 1, 0)} Explanation

Explanation

4. Consider the vector space

R

² over

R

. Then which of the following sets cannot span

R

²?

a:

S

={(1, 4), (2, -8)} Explanation

Explanation

b:

S

={(1, 2), (0, 0), (3, 6)} Explanation

Explanation

c:

S

={(-3, 4), (6, 7)} Explanation

Explanation

d:

S

={(2, 5), (0, 0)} Explanation

Explanation

5. Consider the vector space

R

² over

R

. Then a non-trivial linear combination of elements of which of the following sets equals zero?

a:

S

={(-2, 3), (1, 4)} Explanation

Explanation

b:

S

={(-2, 0), (0, 4), (0, 0)} Explanation

Explanation

c:

S

={(3, 1), (6, 2)} Explanation

Explanation

d:

S

={(2, 2), (6, 6)} Explanation

Explanation

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