3. Classify each of the following statements as always true, sometimes true, or

Question

3. Classify each of the following statements as always true, sometimes true, or never true. No justification required a. If dim(V)-3, and v,.V2V. V4 are vectors in V, then (v,. v2. V,. v4) is a linearly independent set. b. If dim() 3, and vi,V2 are vectors in V, then {v,, V2) is a linearly independent set c. Îf dim(V) = 3 , and v1.v.v3 , v4are vectors in V, then(v1.v..vyN4} spans V. d. If dimV)-3, and (v, V2- V,) is a linearly independent set of vectors in V, then v, v, vj) spans V e. If {y,vN3} is a basis for V, then {3y,y,,-2v3} is also a basis for V. f. If v,V2 V is a basis for V, then (vi + v2. v2 - v,) is a basis for V.

Explanation / Answer

a) NEVER TRUE                  ( because any L.I. set V contains atmost three vectors )

b) SOMETIMES TRUE        ( v1 and v2 may be L.I. or may be L.D. )

c) SOMETIMES TRUE        ( any three vectos from set {v1, v2, v3, v4} may be L.I. or may be L.D. )

d) ALWAYS TRUE             ( set {v1, v2, v3} is L.I. and dim(V) = 3, so set {v1, v2, v3} forms a basis)

e) ALWAYS TRUE

f) NEVER TRUE               ( set {v1+v2, v2-v3} contains 2 vectors which is less than dim(V) }

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