6. Find the standard matrix for the linear transformation T from R³ to R4 if T(ē₁)= [0 1 2 3] T T(ē2) = [4 3 1 7] T T(3) = [250 -1]T and find T([327] T).

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A: Since you have posted multiple question in this question which are not interlinked so i have solved…

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Q: = { [" -d] | a,bER} a X = Prove that and X* are isomorphic

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Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

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Q: find the'mters ection d Exe sy,z]= [a,0,31++ 4,-2,178 [+sy,z]=[6,-12,--11]+5 [1,2,4]?

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A: If A ∈ Rnxm then we have to prove that rank(A) = rank(AT)

Q: < Use exact numbers. y = ←++ x+ -9-8-7-6-5-4-3-2 9 8 7 23456 6- 5- 4+ 3 2- 1- ST -4 -3- -5- Y -6- -7…

A: from given graph two points on line are (-2,0)  and (4,1)standard form of line is y=mx+cwhere m is…

Q: is R* is contractible?

A: This is a problem of Topology.

Q: Q1/ Let f, = (! ? 3 45 145 3 ). = (2 6 5).f = (1 345 6 2) € p. Find the following: 1) f, of, 2) 5 3)…

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Q: B) Show that coth-x = In ()

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Q: In(3*+2*) Apply L'Hospital's theorem to evaluate Lt X00

A: Here we will solve the given limit by L'Hospital Theorem,

Q: +Ex@5)a letfixy befunckien and ASx Bsyshaw tuat %3D ১ ) = AB

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Q: Find the deriratire of y tanx usim the Difinition Rule

A: Given that:y=tanxTo find its derivative.

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Q: 5. Find A-1 and A4 if 3 0 0 0 0-1 -1 0 0 A = 0 0 1/2 0 .0 00 1

A: 5. Given that the matrix is                                To find

Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: As PQR in sicle Circl 2²4²25 Such that & & Rone (3,4) &(-4,3) fmd <Q PR?

A: Given that PQR is a triangle, which is inside the circle  such that P and Q are (-4,3) and (3,4)…

Q: 1. v₁ = (4, −2, 6, −4), V₂ = (6, −3, 9, −6)

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Q: & 18) Vaing showj that the idently in

A: Using the identity 1+x2n=1+xn1+xn,                . . . (1) we want to show that 2nn=∑k=0nnk2.…

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Q: (-8 1 4 (a) If A = 4 4 7 prove that A- = A' where A' is the transpose of A. 1 -8 4

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Q: To which indeterminate forms does l’Hôpital’s Rule applydirectly?

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