Problem 5. Given V1, v3,. . . ,Vm R", let R Rm be defined by: where x, y is the standard inner product of IR Which of the following statement is incorrect?
1. Taking the standard bases Un on R: codomain: 2. Taking the standard bases Un on R": codomain: V2 Mato,--U"' (f) 3. f is a linear transformation. V1 V2 Vm
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Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y)...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y) is the standard inner product of Rn Which of the following statement is incorrect? 1. Taking the standard bases Un on R": codomain: MatUn→Un(f)-(v1 2. Taking the standard bases Un on R: codomain: v2 vm) Matf)- 3. f is a linear transformation. 4. Kerf- x E Rn : Vx = 0 , where: Problem 8. Which of the following statements...
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise add...
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise addition and scalar multiplication of functions. In other words, for f, g E D and λ E R the function R defined by: (f +Ag) ()-f(r) +Ag(x) Let R be four functions defined by: s(x)-: sin 11 c(r) : cosz, co(z)--cos(z + θ), and so(r) sin(z + θ), and Wspanls, c Which of the following statements are true: (a) For each fixed θ...
5. (d) only Problem 4. Let ge,(R) Palarmult plication, and:mer-KS2-beeR} be the vector space of 2...
5. (d) only Problem 4. Let ge,(R) Palarmult plication, and:mer-KS2-beeR} be the vector space of 2 x 2 square matrices with usual matrix addition and State the incorrect statement from the following five: 1. W is a subspace of GE2(R) with basis: of (10 (0 1 (0 0 1-1 0) o 1 2. W Ker f, where GLa(R) 4 R is the linear transformation defined by 3. Given the basis B in option 1., coordB((-2 4. gL2(R) W+V, where: 3(22)...
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ...,...
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,...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
Ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric ...
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is:
ui l uentical...
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R",...
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...
Let x = [xı x2 x3], and let TER → R be the linear transformation defined...
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
please help me with questions 1，2，3 1. Let V be a 2-dimensional vector space with basis...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y) is the standard inner product of Rn Which of the following statement is incorrect? 1. Taking the standard bases Un on R": codomain: MatUn→Un(f)-(v1 2. Taking the standard bases Un on R: codomain: v2 vm) Matf)- 3. f is a linear transformation. 4. Kerf- x E Rn : Vx = 0 , where: Problem 8. Which of the following statements...
Problem 3. Let D be the vector space of all differentiable function R wth the usual pointwise addition and scalar multiplication of functions. In other words, for f, g E D and λ E R the function R defined by: (f +Ag) ()-f(r) +Ag(x) Let R be four functions defined by: s(x)-: sin 11 c(r) : cosz, co(z)--cos(z + θ), and so(r) sin(z + θ), and Wspanls, c Which of the following statements are true: (a) For each fixed θ...
5. (d) only Problem 4. Let ge,(R) Palarmult plication, and:mer-KS2-beeR} be the vector space of 2 x 2 square matrices with usual matrix addition and State the incorrect statement from the following five: 1. W is a subspace of GE2(R) with basis: of (10 (0 1 (0 0 1-1 0) o 1 2. W Ker f, where GLa(R) 4 R is the linear transformation defined by 3. Given the basis B in option 1., coordB((-2 4. gL2(R) W+V, where: 3(22)...
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,...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is:
ui l uentical...
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...
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
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