R³ R³ We consider the linear map f(x,y,z)+(-x+y+z,x-y+z,x-y-32) (1) Determine the kernel of f. (2) Is f injective? Bijective? (3) Write the matrix B of f in the canonical basis of R³.

Q: A linear transformation from Pn to Pn−1, where Pn denotes the space of polynomials of degree not…

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Q: Find a basis for the null space of the matrix A -16 -16 20 -8 -8 -8 10 -4 4 4 -5 2 Entor your basic…

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Q: (b) Consider a linear map ƒ: R³ R³3 and suppose the matrix associated with f under the standard…

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Q: Explain why S is not a basis for R². s={(6, 8), (1, 0), (0, 1)) S is linearly dependent. OS does not…

A: We will use S to determine which among the given options is correct.

Q: Let t be the linear transformation t: R3 (x, y, z) → > R² (3x + y − z, 6x + 2y — 2z). (a) Find a…

A: ### Part (a): Basis for   ImtTo find a basis for the image of t, we need to determine the outputs of…

Q: Check if set X is linearly i ndependent, spans r3 and check for basis of r3 X={(1,0,0),(0,1,0),…

A: A set is a BASIS if it linearly independent and spans the vector space

Q: Consider the bilinear form H: R2 × R2 R defined by →> = x1 y1 + 6x1y2+6x2Y1 + 3x2Y2. H Let B= =…

A: Given the bilinear form  defined by.The ordered basis of R2 is .

Q: Under what conditions on the scalar 'a', do the vectors (1, 1, 1) and (1, a, a²) form a basis of…

A: In order for the vectors (1, 1, 1) and (1, a, a²) to form a basis for C³(C), they must be linearly…

Q: Let B = (v₁.....V₁) be a linearly independent set in R". Explain why B must be a basis for R".

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Q: Explain why S is not a basis for R³. S = {(1, 1, 1), (1,1,0), (1,0,1), (0, 0, 0)) OS is linearly…

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Q: Find the coordinate vector [p(x)]B of p(x) = x² + 4x + 4 with respect to the basis B = {x³ + x – 2,…

A: According to the given information, it is required to find the coordinate vector.

Q: If vi, v2, ..., Vn are independent vectors in R", then they must form a basis of R". True False…

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Q: (c) Let f: R→ R such that f(x, y, z) = (2x - 3y, 2y3z, 2z - 3x). Show that f is a linear…

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Q: 5. ) Let P3 := {ao + a₁x + a2x² + a3x³|ao, a1, a2, a3 € R} be the vector space of polynomials of…

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Q: The coordinates of p(x)=5+3x+x² relative to the basis {1+x,2-x2,3+x+x²} is O (-1,1,2) none (0,1,1) O…

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Q: 2.) Let Tv = Av represent the linear transformation T: R2 → R3 where - 1 21 A = -2 4 L-2 2. Find a…

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Q: M2r2 is the vector space consisting of all 2x2 real matrices. The linear transformation T: M2r2 x →…

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Q: = {(: ") e v : a,b,c, € F}, w. = { (; ") : a, 6.c € F}- a b { (; 2) : a,6, c € F}. Find the basis…

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Q: (a) Given S={(1,-5,4), (11,6,-1), (2,3,5)} the set of vectors in R³. Determine whether (i) S spans…

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Q: 8- 3 3 (a) Find the coordinate vector of a = of R³. (b) Let F₁ be the ordered basis of R2 given by…

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Q: Be S = {v1 = (1, 2, 1), v2 = (2,9, 0), v3 = (3, 3,4)} a subset of R and T: R → R? T(v1) = (1,0),…

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Q: Explain why S is not a basis for R2. S = {(-6, 6)} O sis linearly dependent. S does not span R². S…

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Q: Let B = {v1, v2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B• (a) Is (, )…

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Q: (a) Show that the vectors v₁ = (1,-1,2), v₂ = (2,-1,3) and v3 = (3,c + 1,2 c) do not span R³ and…

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