Question

Q10 10 Points Please answer the below questions. Q10.1 4 Points Let m, n EN\{1}, V be a vector space over R of dimension n and (v1,..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vy) spans V. If (01,..., Vm) is linearly independent then m <n. (V1,..., Um) is linearly dependent if and only if for all i = 1,..., m we have that Vi Espan(v1,..., Vi-1, Vi+1,..., Vyn). Assume there exists exactly one linear transformation T:V + V such that T(vi) Vi for all 1 Sism. Then (V1, ..., Um) is a basis for V. Save Answer

Answer 1

(10.1)

(i) is FALSE because size does not matter here we can take a single vector m times.

(ii) TRUE

because dim(V)=n that means a linearly independent set can have maximum n elements (vectors).

(iii) FALSE

for dependent we need one such i i may not be hold for all i

(iv) FALSE

If we take m<n then it cannot be basis for V.

(10.2)

(i) TRUE

This is the definition of V no need to be surjective it always holds.

(ii) TRUE

Because T is injective which implies both V and W have same dimensions so V0 and W0

(iii) TRUE

I tried to find example to make it false but couldn't you should try atleast once.

(10.3)

(i) FALSE

Not even a subset how can be talk about subspace.

(ii) TRUE

if not basis then either A is not Invertible or given set is not independent.

(iii) FALSE

As dim (R^2)=2 so any set having more than two vectors will necessarily be linearly Dependent.