Linear Algebra Answers

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

Use the properties of determinants to evaluate the following determinant:

(b+c)2 a2 a2
b2 (c+a)2 b2
c2 c2 (a+b)2

Let B1={[1 1], [1 0], [1 0] and B2={[1 0], [0 1], [0 1]
[0 1] [0 1] [0 -1] } [0 0] [0 1] [0 -1] }
be 2 bases for span(B1) in M22, with the usual left to right ordering.
Let B3 be a basis for P1 and be the transition matrix from B2 to B3 giiven by [1 1 1]
[0 1 1] = PB2→B3
[0 0 1]
a) Find transition matrix PB1→B3
b) Use PB2→B3 to find B3

Standard basis vectors for R^3 are (1,0,0),(0,1,0) and (0,0,1). If we want to insert u → into this basic, then which vector from standard basis can be removed while still maintaining the basis of R^3.
Discuss the case when:
u → =(4,3,6)
u → =(4,0,6)
Interpret the result geometrically in both cases.

which sets are a basis for the following vector subspace of P2 :
X={A e M22 : A [1] = [0] }
[2] [0]

A {[2 0] , [0 -1] , [0 0] , [0 0] } C {[2 -1] , [0 0] }
[0 0] [0 0] [2 0] [0 -1] [0 0] [2 -1]

B{[2 -1] } D { [2 -1] , [2 -1] }
[2 -1] [2 -1] [-2 1]