# isomorphism

See the attached file.

1. Find a basis B of R^n such that the B-matrix B of the given linear transformation T is diagonal.

Reflection T about the line in R^n spanned by [2;3

2. Which of the subsets of P_2 given below are subspaces of P_2? Find the basis for those that are subspaces.

i) {p(t):p(0)=2}

ii) {p(t):p(2)=0}

iii) {p(t):p'(1)=p(2)}

iv) {p(t):∫1,0p(t)dt=0}

v) {p(t):p(-t)=-p(t), for all t}

3. Which of the subsets of R^(3×3) given below are subspaces of R^(3×3)?

i) The invertible 3×3 matrices

ii) The diagonal 3×3 matrices

iii) The upper triangular 3×3 matrices

iv) The 3×3 matrices whose entries are all greater than or equal to zero

v) The 3×3 matrices A such that vector [1; 2; 3] is in the kernel of A

vi) The 3×3 matrices in reduced row-echelon form.

4. Find a basis for the spaces and determine its dimension.

i) The space of all 2X2 matrices A such that A[1,1;1,1]=[0.0;0.0]

5. Find all the solutions of the differential equation f^'' (x)-8f^' (x)-20f(x)=0.

6. Make up a second-order linear DE whose solution space is spanned by the functions e^(-x)and e^(-5x).

7. Find out which of the transformations below are linear. For those that are linear, determine whether they are isomorphisms.

i) T(M)=M+I2 from R^(2×2) to R^(2×2)

ii) T(M)=7M from R^(2×2) to R^(2×2)

iii) T(M)=det(M) from R^(2×2) to R

iv) T(M)=M[1,2;3,4] R^(2×2) to R^(2×2)

v) T(M)=S^-1MS, where S=[3,4; 5,6], from R^(2×2) to R^(2×2)

vi) T(M)=PMP-1, where P= [2,3;5,7], from R^(2×2) to R^(2×2)

vii)T(M)=PMQ, where P= [2,3;5,7], and Q= [3,5;7,11], from R^(2×2) to R^(2×2)

viii) T(c)= c[2,3;4,5] from R^(2×2) to R^(2×2)

ix) T(M)=M[1,2;0,1]-[1,2;0,1] M from R^(2×2) to R^(2×2).

https://brainmass.com/math/linear-algebra/linear-spaces-isomorphism-435166

#### Solution Preview

See the attachment for full solution.

1. Let be the reflection about the line spanned by . The equation of this line is . A basis is a sought-for basis if and only if and for some real and . The only points of such that are whose ones which lie at either this line itself or on it's perpendicular line . Therefore the sought-for basis is given, for instance, by points and .The matrix of in this basis is .

2. i) It is not a subspace, since the zero polynomial does not belong it.

ii) It is a subspace, since it is closed under the addition and the product by a scalar. To find its basis, let us note that it is isomorphic to the following subspace of : . It is a plane passing through . It is a 2-dimentional subspace and its basis is given , for instance, by the points (0,-2,1) and (-2,1,0). Therefore the example of basis for the given space of polynomials is given by the polynomials corresponding to the indicated points. These polynomials are and .

iii) It is a subspace, since it is closed under the addition and the product by a scalar. To find its basis, consider any polynomial from . We have . Besides, . Hence . From we obtain

.

Therefore

. (1)

Similarly to ii), it is ...

#### Solution Summary

An isomorphism is depicted in the solution