Linear Algebra Answers

Write the vector (1, −2, 5) as a linear combination of the vectors (1, 1, 1),(1, 2, 3)

and (2, −1, 1)

Consider the following systemof linear equations:

x+2y+2z=1,x+ay+3z=3,x+11y+az=b.

For which values of a does the system have a unique solution. and for which pairs of values (a,b) does the system have more than on solution?

Given that M is a singular matrix, evaluate x where;

6 7 -1

M = 3 x 5

9 11 x

For p∈P3(R) given by p(x)=a0+a1x+⋯+a3x3, let s(p)=a0+a1+a2+a3 and det(p)=a0. Also, corresponding to the polynomial p∈P3(R), we define the polynomial p∗ to be p(−x). Which of the following are subspaces of P3(R) ?

Find the conical form of a quadratic form Q(x,y)=2x^2+2y^2-2xy by using an orthogonal transformation hence find nature,rank,index and signature of the conical form?

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?