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C3 Let Vị and V2 be orthogonal vectors in R”. Prove that ||71 + 72|12 =...
חו (1 point) Suppose V1, V2, V3 is an orthogonal set of vectors in R Let w be a vector in span(V1, V2, V3) such that (v1,vi) = 24, (v2,v2) = 21, (V3, V3) = 9, (w,v) 120, (w, v2) = 147, (w,v3) -36, Vi+ V2+ then w= V3.
(1 point) Suppose V1, V2, U3 is an orthogonal set of vectors in R. Let w be a vector in Span(V1, U2, U3) such that 01.01 = 33, U2 · U2 = 10.25, 03 · 03 = 36, W • V1 = 99, w · U2 = 71.75, w · Uz = -108, then w = Vi+ U2+ U3
Exercise 2.20 Let S= {v, V2, ... , Vpf be a subset of R" containing n vectors. Prove that if S generates R and S is linearly independent, then n m. Exercise 2.20 Let S= {v, V2, ... , Vpf be a subset of R" containing n vectors. Prove that if S generates R and S is linearly independent, then n m.
Let E be the plane in R3 spanned by the orthogonal vectors v1=(121)and v2=(−11−1) The reflection across E is the linear transformation R:R3→R3 defined by the formula R(x) = 2 projE(x)−x (a) Compute R(x) for x=(1260) (b) Find the eigenspace of R corresponding to the eigenvalue 1. That is, find the set of all vectors x for which R(x) =x. Justify your answer.
Suppose V1, V2, V3 is an orthogonal set of vectors in R5. Let w be a vector in span(V1, V2, V3) such that (V1, V1) = 51, (V2, V2) = 638, (V3, V3) = 36, (w, V1) = 153, (w, v2) = 4466, (w, V3) = -36, then W = _______ V1 + _______ V2+ _______ V3.
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R", and let T:R" → R" be a linear transformation such that the set {T(01), T(V2), ...,T(Un) } is basis for R". Show that B = {01, V2, ..., Un } is also a basis for R". Problem 7: Decide whether the following statement is true or false. If it is true, prove it. If it is false, give an example to show that it...
3. Let ū and ū be vectors. Prove that ū x ū is orthogonal to both ū and v.
3. Given pairwise orthogonal vectors u, v, w ER(each vector is orthogonal to every other), with || || = ||0|| = ||w|| = 1, and C1, C2, C3 € R, prove that || Cu + c2v + c3w||2 = cſ + cx+cz.
In R. let V be the orthogonal complement of the vectors u and v, where u = (1,9, 3,61) and v= (4, 36, 13, 254) Find a basis B = {b1,b2} for V: b = 1 Now find five vectors in V such that no two of them are parallel e- LLL
(3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A- (3 points) Let W be the subspace of R spanned by the vectors 1and 5 Find the matrix A of the orthogonal projection onto W A-