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 subset of Rm.

Expert Solution

Colby Messinger answered 2 years ago

Let T : Rn -> Rm be a onto linear transformation. Select all the true statements...

Let T : Rn -> Rm be a onto linear transformation. Select all the true statements from the following list. (Incorrect choices will earn you negative points) For any y in Rm there exists a unique x in Rn such that T(x) = y For any y in Rm there is at most one x in Rn such that T(x) = y The range of T is the entire Rm For any y in Rm there is at least one...

if {v1,v2,v3} is a linearly independent set of vectors, then {v1,v2,v3,v4} is too.

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

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) =...

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) = (2,3,-5) and T( e⃗2 ) = (-1,0,1). Determine the standard matrix of T. Calculate T( ⃗u ), the image of ⃗u=(4,2) under T. Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the image of ⃗w=2⃗u−v⃗ . 4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.

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?

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 : R n → R m be a linear transformation and A the...

. Let T : R n → R m be a linear transformation and A the standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the set BR = {T( r~u +1), . . . , T( ~un)} is a...

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

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