Math, asked by sunitamallik, 1 month ago

assertion(A):if x2 +1/x2=62, then x+1/x=8 reasons (R) :(a+b)2 =a2+b2+2ab​

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Answered by chaurasiaaradhya60
0

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

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