let l be the linear transformation from a vector space V where ker(L)=0 if { v1,v2,v3}...

let l be the linear transformation from a vector space V where ker(L)=0 if { v1,v2,v3} are linearly independent vectors on V prove {Lv1,Lv2,Lv3} are linearly independent vectors in V

Expert Solution

Colby Messinger answered 2 years ago

Let {v1, v2, v3} be a basis for a vector space V , and suppose that...

Let {v1, v2, v3} be a basis for a vector space V , and suppose that w = 3v1 − 5v2 + 0v3. For each of the following sets, indicate if it is: a basis for V , a linearly independent set, or a linearly dependent set. (a) {w, v2, v3} (b) {v1, w} (c) {v1, v2, w} (d) {v1, w, v3} (e) {v1, v2, v3, w}

If {v1, v2, v3, v4} is a linearly-independent subset of a vector space V over the...

If {v1, v2, v3, v4} is a linearly-independent subset of a vector space V over the field Q, is the set {3v1 + 2v2 + v3 + v4, 2v1 + 5v2, 3v3 + 2v4, 3v1 + 4v2 + 2v3 + 3v4} linearly independent as well?

(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2 (b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where...

(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2 (b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where we write V1+V2 to be the subspace of V spanned by V1 and V2 .

v1=[0,1,4] v2=[-4,-5,7] v3=[14,10,8] b=[16,18,19]. Let v1,v2, and v3 be three nonzero vectors in R3. Suppose v2...

v1=[0,1,4] v2=[-4,-5,7] v3=[14,10,8] b=[16,18,19]. Let v1,v2, and v3 be three nonzero vectors in R3. Suppose v2 is not a scalar multiple of either v1 or v3 and v3 is not a scalar multiple of either v1 or v2. Does it follow that every vector in R3 is in span{v1,v2,v3}?

Let T : Rn →Rm be a linear transformation. (a) If {v1,v2,...,vk} is a linearly dependent...

Let T : Rn →Rm be a linear transformation. (a) If {v1,v2,...,vk} is a linearly dependent subset of Rn, prove that {T(v1),T(v2),...,T(vk)} is a linearly dependent subset of Rm. (b) Suppose the kernel of T is {0}. (Recall that the kernel of a linear transformation T : Rn → Rm is the set of all x ∈ Rn such that T(x) = 0). If {w1,w2,...,wp} is a linearly independent subset of Rn, then show that {T(w1),T(w2),...,T(wp)} is a linearly independent...

Let v1 = [-0.5 , v2 = [0.5 , and v3 = [-0.5 -0.5 -0.5 ...

Let v1 = [-0.5 , v2 = [0.5 , and v3 = [-0.5 -0.5 -0.5 0.5 0.5 0.5 0.5 -0.5] 0.5] 0.5] Find a vector v4 in R4 such that the vectors v1, v2, v3, and v4 are orthonormal.

We are given a set of vectors S = {V1, V2, V3} in R 3 where...

We are given a set of vectors S = {V1, V2, V3} in R 3 where eV1 = [ 2 −1 3 ] , eV2 = [ 5 7 −1 ] , eV3 = [ −4 2 9 ] Problem 1 • Prove that S is a basis for R^3 . • Using the above coordinate vectors, find the base transition matrix eTS from the basis S to the standard basis e. Problem 2 Using your answers in Problem 1...

Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V...

Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V and Φ (V) is a subspace of W.

Let G a graph of order 8 with V (G) = {v1, v2, . . ....

Let G a graph of order 8 with V (G) = {v1, v2, . . . , v8} such that deg vi = i for 1 ≤ i ≤ 7. What is deg v8? Justify your answer. Please show all steps thank you

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