assertion(A):if x2 +1/x2=62, then x+1/x=8 reasons (R) :(a+b)2 =a2+b2+2ab
Answers
Answer:
6. Let Pk be the space of polynomials of degree at most k and define the linear map
L : Pk → Pk+1 by Lp := p
00(x) + xp(x).
a) Show that the polynomial q(x) = 1 is not in the image of L. [Suggestion: Try
the case k = 2 first.]
b) Let V = {q(x) ∈ Pk+1 | q(0) = 0}. Show that the map L : Pk → V is invertible.
[Again, try k = 2 first.]
7. Compute the dimension and find bases for the following linear spaces.
a) Real anti-symmetric 4 × 4 matrices.
b) Quartic polynomials p with the property that p(2) = 0 and p(3) = 0.
c) Cubic polynomials p(x, y) in two real variables with the properties: p(0, 0) = 0,
p(1, 0) = 0 and p(0, 1) = 0.
d) The space of linear maps L : R
5 → R
3 whose kernels contain (0, 2, −3, 0, 1).
8. a) Compute the dimension of the intersection of the following two planes in R
3
x + 2y − z = 0, 3x − 3y + z = 0.
b) A map L : R
3 → R
2
is defined by the matrix L :=
1 2 −1
3 −3 1
. Find the
nullspace (kernel) of L.
9. If A is a 5 × 5 matrix with det A = −1, compute det(−2A).
10. Does an 8 -dimensional vector space contain linear subspaces V1 , V2 , V3 with no com-
mon non-zero element, such that
a). dim(Vi) = 5, i = 1, 2, 3? b). dim(Vi) = 6, i = 1, 2, 3?
11. Let U and V both be two-dimensional subspaces of R
5
, and let W = U ∩ V . Find all
possible values for the dimension of W .
12. Let U and V both be two-dimensional subspaces of R
5
, and define the set W := U +V
as the set of all vectors w = u + v where u ∈ U and v ∈ V can be any vectors.
a) Show that W is a linear space.
b) Find all possible values for the dimension of W .
13. Let A be an n×n matrix of real or complex numbers. Which of the following statements