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The transpose of matrix (begin{bmatrix}1&0&-7 0&-2&3 4&5&6 end{bmatrix}) a.(begin{bmatrix}1&0&4 0&-2&5 -7&3&6 end{bmatrix}) b.(-83) c.(-59) d.(begin{bmatrix}-1&0&-4 0&2&-5 7&-3&-6 end{bmatrix})
If V = P3 with the inner product < f, g >= R 1 11 f(x)g(x)dx, apply the Gram-Schmidt algorithm to obtain an orthogonal basis from B = {1, x, x2 , x3}
Let V = M2×w(R) and W = P2(R). Define T  a b c d = a + b + (c − d)x + bx2 . Let β = 1 0 0 0 ,  0 1 0 0 ,  0 0 1 0 ,  0 0 0 1 and γ = {1, x, x2 }. Find φβ and φγ. Find the matrix A so that LAφβ = φγT. Support your answer by evaluating both maps on M =  3 1 −1 4
Suppose T : V → V is linear and let W be a subspace of V . Further suppose that T(w) ∈ W for all w ∈ W. Let S : W → W be defined by S(w) = T(w). (a) Find an example of such a T and {0} 6= W 6= V when V = R 2 . (b) Prove (in general) that N(S) = N(T) ∩ W.
Let V be the vector space of all sequences over R. Given (a1, a2, . . .) ∈ V , define T, U : V → V by T(a1, a2, a3, a4, . . .) = (a1, a3, a5, . . .) and U(a1, a2, a3, a4, . . .) = (0, a1, 0, a2, 0, a3, . . .) (a) Find N(T) and N(U). (b) Explain why T is onto, but not 1-1. (c) Explain why U is 1-1, but not onto.
Let L = {(1, 1, 1, 1, −4),(1, −1, 3, −2, −1)}. Find 6 vectors in the collection, say H, such that L ∪ H spans the entire space.
Check p(x) + p(−x) ∈ P (e) for every p(x) ∈ R(x). Check that the map ψ : R[x] → P (e) given by ψ(p(x)) = p(x)+p(−x) 2 is a linear map. Further, check that ψ 2 = ψ. Determine the kernel of ψ.
given that\\(A=\\begin{pmatrix}1 & 2 & 3\\\\ 4 & 5 & 6 \\end{pmatrix}\\)\nand \\[B=\\begin{pmatrix} 1 & 2\\\\ 3& 4\\\\ 5& 6 \\end{pmatrix}\\]\n. Find AB
Determine if the matrix p= [√3/3, √6/6, -√2/2 ] [-√3/3, √6/3, 0 ] [√3/3, √6/6, √2/2] is othognal.
Determine if the matrix q= √3/3, √6/6, -√2/2 -√3/3, √6/3, 0 √3/3, √6/6, √2/2 Is othognal.
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