Linear Algebra Answers

Let A={b1,b2,b3} be a set of three-dimensional vectors in R3.
a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.
b. Prove that if the set A spans R3, then A is a basis of R3.

QUESTION 5

Let a11 x1 + a12 x2 + a13x3 = b1

a21 x1 + a22 x2 + a23 x3 = b2

a31 x1 + a32 x2 + a33 x3 = b3.

Show that if det (A) ≠ 0 where det(A) is the determinant of the coefficient matrix;

then x2 = det(A2)/det(A) where det(A2) is the determinant obtained by replacing the second column of det(A)

by (b1; b2; b3)^T

QUESTION 5

Let a11 x1 + a12 x2 + a13x3 = b1

a21 x1 + a22 x2 + a23 x3 = b2

a31 x1 + a32 x2 + a33 x3 = b3.

Show that if det (A) ≠ 0 where det(A) is the determinant of the coefficient matrix;

then x2 = det(A2)/det(A) where det(A2) is the determinant obtained by replacing the second column of det(A)

by (b1; b2; b3)^T

2. Let A={b1,b2,b3} be a set of three-dimensional vectors in R3.
a. Prove that if the set A is linearly independent, then A is a basis of the vector space R3.

[|111||230||383|] ,is the matrix invertible?

can the system
3xy-4y-z=5
3x+2y-z=0
x-y+z=1
be solved by using Cramer's rule? if not, why? if yes, solve it.

Now suppose that A and B are arbitrary n  n matrices such that A and 2A + BT are invertible.
Using general properties of matrix operations show
that (A-1BT + 2I)-1 = (2A + BT)-1A. Show all the steps.

Given z=cos A + i sin A and u+iv =(1+z)(1+z^2).Prove that v=u tan (3A/2),u^2+ v^2=16 cos^2(A/2)cos^2(A).Hint:A=theta

consider the vectors;u(1,0)and v(0,1)
questions
1-determine cos theta,where theta is the angle between u and v.
2-determine the area of the parallelogram determined by u and v.


question 2
2.1-Let L1 and L2 be defined by x=W0+su where s is the element of real numbers
and y=w1+tv where t is the element of real numbers .
2.2-find the plane the passes through the point(2,4,-3)and is parallel to the plane -2x+4y-5z+5=0
2.3.find the line that passes through the point (2,5,3)and is perpendicular to the plane 2x-3y+4z+7=0
2.4find an equation of the plane passing through the point(-2,3,4)and is perpendicular to the line passing through the points (4,-2,5)and (0,2,4)