Linear Algebra Answers

consider the following system of equations:

3x + 4y +5z = 66

7x + 4y +3z = 74

8x + 8y +9z = 136

a. write down the associated augmented matrix for this system of equations and the coefficient matrix A.

b. by performing elementary row operations of the augmented matrix, solve the system of equations or show that no solution exists. In case there exist infinitely many solutions, then the solution to the system must be written in parametric form.

c. Based on your answer in b, what is the rank of the coefficient matrix A?

Find the product of eigen value of matrix

Let A, B be two subrings of a ring R such that for all a e A, b e B,

ba e A then show that

(1) A + B is a subring of R.

(2) A is an ideal of A + B.

3) An B is an ideal of B.

Reduce the quadratic form 2𝑥

2 + 2𝑦

2 + 2𝑧

2 + 2𝑦𝑧 to the canonical form by 

orthogonal reduction. Find the index, signature and nature of the quadratic form.

Express v = t2+4t-3 in P(t) as a linear combination of the

polynomials p =t2−2t+5,p =2t2 −3tand p =t+1.

Define the following terms

PART 1.

a) What is an equation?

b) What is a linear equation?

c) What is determinants?

d) What is a matrix?

e) What is linear programming?

Question: Apply the Linear dependence and Linear Independence in vector definitions and show that the given vectors are Linearly dependant or independent vectors in R⁴.

V₁ = (1, 3, -1, 4)^T, V₂= (3, 8, -5, 7)^T, V₃= (2, 9, 4, 23)^T.

Question. Given the matrix A = 3 1 1

2 4 2

1 1 3

a. Solve A for its eigen values and given vectors

b. Constant Similar matrix for A if possible

Question 1. Let u = (2,1,0,5) e R². Apply the process of inner product spaces.

• Solve u for its length
• Solve u for its Normalized vector

Let T

 be a function from R

3

→R

3

 defined by

T(x,y,z)=(x−y+2z,2x+y,−x−2y+2z)

(i)

(ii)

  Show that T is a Linear Transformation

  Find nullity of T