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?

Expert Solution

Colby Messinger answered 2 years ago

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

V1 V2 V3 V4 V1 1.0 V2 .27 1.0 V3 -.13 .65 1.0 V4 .20 -.15...

V1 V2 V3 V4 V1 1.0 V2 .27 1.0 V3 -.13 .65 1.0 V4 .20 -.15 -.72 1.0 IN THIS EXERCISE, YOU WILL SEE A CORRELATION MATRIX. EXAMINE THE MATRIX AND ANSWER THE QUESTIONS THAT FOLLOW. 1. Which two variables have the strongest (largest) relationship? 2. Which two variables have the weakest (smallest) relationship? 3. Which two variables have the strongest positive relationship? 4. which two variables have the stronger negative relationship? 5. Which two variables have the weakest positive...

consider the vectors: v1=(1,1,1) v2=(2,-1,1) v3=(3,0,2) v4=(6,0,4) a)find the dimension and a basis W=Span(v1,v2,v3,v4) b) Does...

consider the vectors: v1=(1,1,1) v2=(2,-1,1) v3=(3,0,2) v4=(6,0,4) a)find the dimension and a basis W=Span(v1,v2,v3,v4) b) Does the vector v=(3,3,1) belong to W. Justify your answer c) Is it true that W=Span(v3,v4)? Justify your answer

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

(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 .

A basis of a vector space V is a maximal linearly independent set of vectors in...

A basis of a vector space V is a maximal linearly independent set of vectors in V . Similarly, one can view it as a minimal spanning set of vectors in V . Prove that any set S ⊆ V spanning a finite-dimensional vector space V contains a basis of V .

Proof: Let S ⊆ V be a subset of a vector space V over F. We...

Proof: Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.

#1 Let H= Span{v1,v2,v3,v4}. For each of the following sets of vectors determine whether H is...

#1 Let H= Span{v1,v2,v3,v4}. For each of the following sets of vectors determine whether H is a line, plane ,or R3. Justify your answers. (a)v1= (1,2,−2),v2= (7,−7,−7),v3= (16,−12,−16),v4= (0,−3,−3) (b)v1= (2,2,2),v2= (6,6,5),v3= (−16,−16,−14),v4= (28,28,24) (c)v1= (−1,3,−3),v2= (0,0,0),v3= (−2,6,−6),v4= (−3,9,−9) #2 Plot the linesL1: x= t[4−1] and L2: x= [−4−2] + t[4−1] using their vector forms. If[12k]is onL2. What is the value of k?

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 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.

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