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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).

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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 ...

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