Linear Algebra Answers

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample

If W1 and W2 are subspaces of vector space V and W1+W2 =V, thenW1∩W2 = {0}

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If {v1, v2, .... , vn} is a basis for vector space V, then {v1+v2+ +vn, v2,.... ,vn} is also
a basis for V

Q.Give some detail explanation on Pseudo inverse matrix???

Let T : R^2 - R^2 and S: R^2 - R^2 be linear operators defined by
T (x1;x2) = (x1+x2, x1-x2) and S(x1;x2) = (x1, x1+2x2)
respectively.
i) Find T ◦ S and S ◦ T.
ii) Let B = f(1, 0), (0, 1) be the standard basis of R^3. Verify that
[T ◦ S]B = [T]B ◦ [S]B.

Let V be the vector space of polynomials with real coefficients and of degree at most 2.
If D = d/dx is the differential operator on V and B ={1+2x^2, x+x^2, x^2} is an ordered basis of V,
find [D]B.
Find the rank and nullity of D.
Is D invertible? Justify your answer.

Check that the vectors u =(3/5, 4/5, 0) v=(-4/5, 3/5, 0)
and w = (0;0;1) are orthonormal.
Further, write the vector a = (1;-1;2) as a linear combination of the vectors.

Let a =(1/2√2, √3/2√2, 1/√2) and b=(1/√2, 0, 1√2)

i) Find the direct cosines of a and b.
ii) Find the angle between a and b.

a) Obtain the solution set of the system x – 3y + 4z = 9, 4x + 3y + 2z = 7, y – 2x = 5 – 10z by
elimination.
b) Express the following problem situation as a linear system, and then solve it by substitution. Also
show the linear system geometrically.
A manufacturer produces two types of cupboards – deluxe and regular. Each deluxe cupboard
requires 12 worker hours to cut and assemble, and 5 worker hours to finish. Each regular
cupboard requires 8 worker hours to cut and assemble, and 3 hours to finish. On a daily basis,
the manufacturer has available 440 worker hours for cutting and assembling, and 175 worker
hours for finishing. How many cupboards of each type should be produced so that all the work
power is utilized?


c) Write the systems obtained in a) and (b) in matrix form.

Obtain the solution set of the system x – 3y + 4z = 9, 4x + 3y + 2z = 7, y – 2x = 5 – 10z by
elimination.

Use the properties of determinants to evaluate the following determinant:

(b+c)^2 a^2 a^2
b^2 (c+a)^2 b^2
c^2 c^2 (a+b)^2