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(1)MA 1111: Linear Algebra I Tutorial problems, October 19, 2018 In problems 1–4, determine whether the system of vectors {vi} in the vector space V (a) is linearly independent

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MA 1111: Linear Algebra I Tutorial problems, October 19, 2018

In problems 1–4, determine whether the system of vectors {vi} in the vector space V (a) is linearly independent; (b) complete; (c) forms a basis.

1. V =R2, v1= −1

1

, v2= 1

1

. 2. V =R2, v1=

−1 1

, v2=

2 1

, v3=

1 1

.

3. V =R3, v1=

 2

−1

−1

, v2=

−1 2

−1

,v3 =

−1

−1 2

, v4 =

 3

−5 2

, v5 =

 3

−1 1

.

4. V =R3, v1=

 0 1 1

, v2 =

 1 0 1

,v3 =

 1 1 0

.

5. (a) Prove that if the system of vectors u, v, w of some vector space is linearly independent, then the system of vectorsu−2w,v+w,wis linearly independent as well.

(b) Prove that if the system of vectors u,v, wof some vector space is complete, then the system of vectors u−2w,v+w, w is complete as well.

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