1. Conrm, complete, or correct the following denitions of the italicized term. Copy the given denition in your answersheet. To conrm clearly write conrmed, to correct clearly cross out the incorrect part, and to complete clearly circlewhat you add.1. Linear transformation from Rn to Rm is a mapping (i.e., a function) satisfying the following:• T (~x + ~y) = T (~x) + T (~y) for all ~x, ~y 2 Rn (that is, T respects addition”).• T (a~x) = aT (~x) for a nonzero a 2 R and ~x 2 Rn (that is, T respects scalar multiplication”).2. A line in Rn. is any set of the formL = ft~m +~b j t 2 Rg3. A plane in Rn is any set of the formΛ = ft~m + s~n +~b j t; s 2 R; t 6= 0; s 6= 0g:4. A linear combination of ~v1; :::; ~vn where ~vi 2 Rn for all i = 1; 2:::n is a vector of the formfr1~v1 + r2~v2 + ::: + rn~vnjri 2 R; gwhere ri are xed real number for all i:5. A linear combination of ~v1; :::; ~vn where ~vi 2 Rn for all i = 1; 2:::n is a set of vectors dened byfr1~v1 + r2~v2 + ::: + rn~vnjri 2 R; 0 ≤ i ≤ ngwhere ri are xed real number for all i:6. If V and W are vector spaces, an isomorphism from V to W is an injective linear transformation from V to W .7. If V and W are vector spaces, an isomorphism from V to W is an surjective linear transformation from V to W .8. A matrix A is called an invertible matrix , if there exists a matrix B such thatAB = BA = I:9. A square matrix A is called an invertible matrix, if there exists a matrix B such thatAB = I:10. We say that vector space V is isomorphic to vector space W if any linear transformation from V to W in anisomorphism.11. The kernel of a linear map T : V -! W is the set of all non-zero vectors ~v 2 W such that T (~v) = ~0:12. The image of a linear map T : V -! W is the set of all vectors ~v 2 V such that T (~v) 6= 0:13. The image of a linear map T : V -! W is the set of all non zero vectors ~u 2 V such that T (~v) = ~u:14. If V is a vector space, a subspace of V is any set which is a closed under addition and scalar multiplication. Thatis, a subspace of V is a set W such that(a) if ~x; ~y 2 W , then also ~x + ~y 2 W ;(b) if ~x 2 W and k is any scalar, then also k~x 2 W .15. If V is a vector space, a subspace of V is a non-empty set W which contains zero the vector and(a) if ~x; ~y 2 V , then also ~x + ~y 2 W ;(b) if ~x 2 V and k is non-zero scalar, then also k~x 2 W .2. State whether each statement is true or false and provide a short justication for your claim (a short proof if you thinkthe statement is true or a counter example if you think it is false).1. For every 2 × 2 matrix A, there exists a 2 × 2 matrix B such that det(A + B) 6= det A + det B.2. Every linear transformation R6 ! R6 is an isomorphism.3. If all the entries of an invertible square matrix A are integers, the same is true for A-1.4. Suppose A is a m × n matrix whose columns span a subspace V of Rm. If ~b 2 V ?, then A~x = ~b has a unique leastsquares solution.5. There is a linear transformation R5 ) R3 sending ~e1 and ~e2 to 2 41 0 23 5 ; and ~e3 to 2 4-1 213 5 :6. There exists a 3 × 3 matrix P such that the linear transformation T : R3×3 ) R3 dened by T(M) = MP – PM isan isomorphism.7. Every surjective linear transformation V ) V is an isomorphism.8. The dierentiation map from the space P of all polynomials to itself is an isomorphism.9. The map of P to itself dened by f 7! xf is an injective linear transformation.10. The map of P to itself dened by f 7! f2 is an injective linear transformation.11. Any four dimensional vector space has innitely many three dimensional subspaces.12. The set of orthogonal n × n matrices is a subspace of Rn×n.13. The set of symmetric n × n matrices is a subspace of Rn×n.14. There is a two-dimensional subspace of R2×2 whose non-zero elements are all invertible.15. Any two vector spaces of dimension six are isomorphic.16. Let P be a plane in Rn. If T : Rn -! Rm is a linear map, then T(P) must be a plane in Rm.17. Let L be a line in Rn. If T : Rn -! Rm is a linear map, then T(L) must be a line in Rm.18. Let P be a plane in Rn. If T : Rn -! Rm is a linear map, then T(P) can not be a line in Rm.19. Let P be a plane in Rn. If T : Rn -! Rm is a linear map, then T(P) can not be a point in Rm.20. Let L be a line in Rn. If T : Rn -! Rm is a linear map, then T(L) can not be a point in Rm.21. Let A; B be two n × n matrices. If AB = C, then BA = C:22. Let A be a 2 × 2 matrix. There exist a matrix B such thatAB = BA = I2:23. Let T : Rn -! Rm be a linear map. There exists a map Q : Rm -! Rn such thatT ◦ Q = Q ◦ T = Id:24. Let A be a 2 × 2 matrix and det(A) 6= 0. There always exist a matrix B such thatAB = BA = I2×2:25. Let T : Rn -! Rn be a bijective linear map. There always exist a map Q : Rn -! Rn such thatT ◦ Q = Q ◦ T = Id:26. Let n > m. There exists an invertible linear map T : Rn -! Rm.27. Empty set is a subspace of any vector spaces.28. f(x; y) 2 R2 j xy = 1g is a subspace of R2:29. f(x; y) 2 R2 j x – y = 1g is a subspace of R2:30. f(x; y) 2 R2 j x2 + y2 = 1g is a subspace of R2:31. f(x; y) 2 R2 j x < 0 and y > 0g is a subspace of R2:3. In each part, give an explicit example of the mathematical object described or explain why such an object doesn’t exist.1. A linear map L : R2 ! R2 that rotates every vector in R2 counter-clockwise direction by 45◦ while xing the origin.2. A linear map L : R2 ! R2 that dilates every vector in R2 by a factor of λ 2 R.3. A linear map L : R2 ! R2 that reects every vector in R2 over the y-axis.4. A linear map L : R2 ! R2 such that ker(L) contains the vector 1 1.5. A linear map L : R2 ! R2 such that im(L) contains the vector 1 1.6. A linear map L : R2 ! R2 such that ker(L) = fab ; a – b = 1g.4. Carefully prove or disprove the following statements.In case the statement is given in words, rst write down what itmeans mathematically. Justify every step.1. A linear map L : Rn ! Rm should always x the origin. (ie: L(~0) = ~0.)2. A bijective linear map L : R3 ! R3 can send a line to a point.3. A bijective linear map L : R3 ! R3 can send a plane to a point.4. A bijective linear map L : R3 ! R3 can send a plane to a line.5. If L : R3 ! R3 is a bijective linear map, then it must send a plane to a plane.6. If L : R3 ! R3 is a bijective linear map, then it must send a line to a line.7. Let TA : R2 ! R2 be the transformation dened by TA(~x) = A~x for all ~x 2 R2. If det(A) = 0 then TA is notinvertible.8. Let TA : R2 ! R2 be the transformation dened by TA(~x) = A~x for all ~x 2 R2. If det(A) 6= 0 then TA is invertible.9. Let TA : R2 ! R2 be the transformation dened by TA(~x) = A~x for all ~x 2 R2. If TA is invertible, then det(A) 6= 0:10. Let TA : R2 ! R2 be the transformation dened by TA(~x) = A~x for all ~x 2 R2. If TA is an isomorphism, thendet(A) 6= 0:11. If a map f : Rn ! Rm is an isomorphism, then m = n:12. Let TA : Rn ! Rn be the transformation dened by TA(~x) = A~x. Prove that if TA is invertible then there exists amatrix B such that AB = BA = In.13. Let A be an n × n matrix. Prove that if there exists a matrix B such that AB = BA = In then the lineartransformation TA : Rn ! Rn dened by TA(~x) = A~x is invertible.14. f~0g is a subspace of Rn:15. If linear map T : V -! W is injective, then ker(T ) = f~0g:16. There exist a injective linear map T : V -! W such that ker(T ) 6= f~0g:17. Let T : V ! W be a linear transformation. Prove that imT is closed under vector addition. That is if ~w1; ~w2 2 imTthen ~w1 + ~w2 2 imT18. Let T : V ! W be a linear transformation. Prove that imT is closed under scalar multiplication. That is if ~w 2 imTand r 2 R then r ~w 2 imT

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