Questions:
1. Find a basis B of ℜ such that the B-matrix B of the given linear transformation T is diagonal.
Reflection T about the line in ℜn spanned by [2]
[3]
2. Which of the subsets of P2 given below are subspaces of P2? 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):∫10 p(t)dt=0}
v) {p(t):p(-t)=-p(t), for all t}
3. Which of the subsets of ℜ3x3 given below are subspaces of ℜ3x3?
i) The invertible 3x3 matrices
ii) The diagonal 3x3 matrices
iii) The upper triangular 3x3 matrices
iv) The 3x3 matrices whose entries are all greater than or equal to zero
v) The 3x3 matrices A such that vector [1] is in the kernel of A
[2]
[3]
vi)The 3x3 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] = [ 0 0]
[ 1 1] [ 0 0]
5. Find all the solutions of the differential equation ƒ" (x) - 8ƒ' (x) -20(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 ℜ2x2 to ℜ2x2
ii) T(M)=7M from ℜ2x2 to ℜ2x2
iii) T(M)=det(M) from ℜ2x2to ℜ
iv) T(M)=M [1 2] ℜ2x2 to ℜ2x2
[3 4]
v) T(M)=S-1MS, where S= [ 3 4] ,from ℜ2x2 to ℜ2x2
[5 6]
vi) T(M)=PMP-1, where P= [ 2 3], ℜ2x2 to ℜ2x2
[5 7]
vii) T(M)=PMQ, where P= [ 2 3] , and Q= [ 3 5] , from to ℜ2x2 to ℜ2x2
[ 7 11]
[ 5 7]
viii)T(c)= c [ 2 3] from ℜ2x2 to ℜ2x2
[ 4 5]
ix)T(M)= [ 1 2] - [ 1 2] M from ℜ2x2 to ℜ2x2
[ 0 1] [ 0 1]