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(Select ALL that are TRUE) R2 is a subspace of R3. Let (v1, ..., Un) be...
Q10 10 Points Please answer the below questions. Q10.1 4 Points Let m, n EN\{1}, V be a vector space over R of dimension n and (v1,..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vy) spans V. If (01,..., Vm) is linearly independent then m <n. (V1,..., Um) is linearly dependent if and only if for all i = 1,..., m we have that Vi Espan(v1,..., Vi-1, Vi+1,...,...
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1, ..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vm) spans V. If (v1, ..., Um) is linearly independent then m <n. (v1, ..., Um) is linearly dependent if and only if for all i = 1,..., m we have that U; Espan(vi, .., Vi-1, Vj+1, ..., Um). Assume there exists exactly one...
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,...
Let {v1, v2,v3} be a linearly independent set in R^n and let v = -αv3 +v1,w = v2 - αv1, u= v3-αv2 where αER, find all the values of α, where v, w, u are linearly dependent. do not use matrices.
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
3. [1 mark each] Determine which of the following statements are true and which are false. (a) The inverse of a rotation matrix (Rº) is (R-8). (b) If the vectors V1, V2, ..., Vk are such that no two of these vectors are scalar multiples of each other then they must form a linearly independent set. (c) The set containing just the zero vector, {0}, is a subspace of R”. (d) If v, w E R3 then span(v, w) must...
(a) (5 points.) Let W CW CW CW3 be distinct subspaces of R? True/False (Justify your answers): (i) Wo must be the zero subspace. (ii) W, must be R. (iii) W, must be RP. (iv) Suppose V1, V2, V3 are vectors such that vi EWW -for each 1 <i<3. Then {V1, V2, V3} must be a basis for R. (v) There are three linearly independent vectors in R that do not form a basis for R?
Let V be a vector space over R and let v1, ..., Un each be a vector in V \{0}. Show that (v1, ..., Un) is linear independent if and only if span(V1, ..., vi) n span(Vi+1, ..., Un) = {0} for all i = 1,...,n - 1
(12 points) Let vi = 1 and let W be the subspace of R* spanned by V, and v. (a) Convert (V. 2) into an ohonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = { (b) Find the projection of b = onto W (c) Find two linearly independent vectors in R* perpendicular to W. Vectors = 1