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linear independence question 20. Let V1, V2, ...,Vn be linearly independent vectors in a vector space...
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
Problem 1. Let V1, V2 be linearly independent vectors in R3. Using definition of linear independence to show that {3v1 +4v2, 4v1 + 7v2} is linearly independent.
(1) Let S (v1, V2,..., Vn be a set of vectors in a vector space V. (a) Define what does it mean for S to be linearly independent. (b) Define what does it mean for S to be a spanning set for V. (c) Define what does it mean for S to be a basis for v.
Let V be a vector space. Suppose dimV = n and {V1, V2, ..., Vn} is a basis of V. Thei which of the following is always true? a) Any set of n vectors is linearly dependent b) Any linearly dependent set in V is not part of basis c) Any linearly dependent set with n - 1 vectors is a basis d) A linearly independent set with n vectors is a basis
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...
suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).
(3) Let V denote a vector space over the field F and let v,..., Un E V. (a) Show that span(vn, 2,. , Un) (b) Show that span (ui , U2 , . . . , vn) span(v)+ +span(vn). span(v1)@span(v2)㊥·..㊥8pan(vn) if and only if (vi , . , . , %) is linearly independent.
Q4 6 Points Let V be a vector space over R and let V1, ... , Vn each be a vector in V \{0}. Show that (v1, ..., Vn) is linear independent if and only if span(v1, ... , Vi) n span(vi+1, ..., Vn) = {0} for all i = 1,...,n - 1
(4) Let {V1, V2, ..., Vn} be a basis for a vector space V. If w is an element of V whose coefficient vector is the zero vector, show that w must be the zero element.
Let v1 and v2 be two arbitrary linearly independent vectors in R^n . Are (v1 + v2) and (v1 − v2) necessarily linearly independent? Justify.