1. Let v1, . . . , vn be nonzero vectors such that each vi+1 has...

1. Let v1, . . . , vn be nonzero vectors such that each vi+1 has more leading 0s than vi . Show that vectors v1, . . . , vn are linearly independent.

Expert Solution

I am giving two different proofs first is by mathematical induction and second by using echleon form property of matrices. Both proofs are correct. I prefer first one proved by mathematical induction. Please feel free to ask for any query and rate positively.

Colby Messinger answered 2 years ago

Let W be an inner product space and v1, . . . , vn a basis...

Let W be an inner product space and v1, . . . , vn a basis of V . Show that <S, T> = <Sv1, T v1> + . . . + <Svn, T vn> for S, T ∈ L(V, W) is an inner product on L(V, W).

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection...

5. Prove the Following: a. Let {v1, . . . , vn} be a finite collection of vectors in a vector space V and suppose that it is not a linearly independent set. i. Show that one can find a vector w ∈ {v1, . . . , vn} such that w ∈ Span(S) for S := {v1, . . . , vn} \ {w}. Conclude that Span(S) = Span(v1, . . . , vn). ii. Suppose T ⊂ {v1,...

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}?

In each case, check that { v1,...vn} is a basis for R^n, and express the given...

In each case, check that { v1,...vn} is a basis for R^n, and express the given vector b as a linear combination of the basis vectors. (a). v1=(2,3), v2=(3,5). b=(3,4) (b) v1=(1,0,3), v2=(1,2,2), v3=(1,3,2). b=(1,1,2) (c) v1=(1,0,1), v2=(1,1,2), v3=(1,1,1). b=(3,0,1)

#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?

Let T,S : V → W be two linear transformations, and suppose B1 = {v1,...,vn} andB2...

Let T,S : V → W be two linear transformations, and suppose B1 = {v1,...,vn} andB2 = {w1,...,wm} are bases of V and W, respectively. (c) Show that the vector spaces L(V,W) and Matm×n(F) are isomorphic. (Hint: the function MB1,B2 : L(V,W) → Matm×n(F) is linear by (a) and (b). Show that it is a bijection. A linear transformation is uniquely specified by its action on a basis.) need clearly proof

a, The vectors v1 = < 0, 2, 1 >, v2 = < 1, 1, 1...

a, The vectors v1 = < 0, 2, 1 >, v2 = < 1, 1, 1 > , v3 = < 1, 2, 3 > , v4 = < -2, -4, 2 > and v5 = < 3, -2, 2 > generate R^3 (you can assume this). Find a subset of {v1, v2, v3, v4, v5} that forms a basis for R^3. b. v1 = < 1, 0, 0 > , v2 = < 1, 1, 0 > and v3...

Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose...

Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose that v1; v2 are linearly independent and that w1;w2;w3 span W. (a) If dimW = 3 prove that there is a vector in W that is not equal to a linear combination of v1 and v2. (b) If w3 is a linear combination of w1 and w2 prove that w1 and w2 span W. (c) If w3 is a linear combination of w1 and...

Are the vectors v1 = (1 , 2, 3), v2 = (2, 4, 6), and v3...

Are the vectors v1 = (1 , 2, 3), v2 = (2, 4, 6), and v3 = (1, 1, 3) linearly independent or dependent? Since v2 is a scalar multiple of v1, both v1 and v2 are linearly dependent, but what does that say about the linear dependence of the three vectors as a whole?

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

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