Linear Algebra (Part 2)

Problem 1

Solve the system of linear equations 5x-4y=1, -3x+2y=7 over the real numbers using the row reduction approach, the inverse approach, and Cramer’s Rule.

Problem 2

Solve the system of linear equations 2x+3y=8, 4x+7y=1 over (Subscript[ℤ, 11]) using the inverse approach and by Cramer’s rule.

Problem 3

Find all values of a and b for which the system x+2y+3z=1, 2x+y+2z=3, -x +2y+ a z=b has infinitely many solutions.

Problem 4

Find the value of y in the solution to 3x-y+z=1, 2x+2y+z=5, 5x+y-z=0.

Problem 5

Determine if the vectors <1,3,4,1>, <2,1,1,1>, and <-2,-9,-13,1> are linearly independent in (ℝ^4).

Problem 6

Determine if the vectors (({{2}, {3}, {1}})), (({{3}, {1}, {1}})), and (({{4}, {13}, {3}})) span (ℝ^3).

Problem 7

Show that the vectors <2,1> and <3,5> are a basis for (ℝ^2).  Find the coordinates of the vector <1,2> with respect to this basis.

Problem 8

Let V be the span of the vectors <1,2,3,4,1>, <2,1,-1,2,3>, <0,0,0,0,0>, <4,5,5,10,5>, and <7,5,0,10,10>  in (ℝ^5). What is dim(V)?

Problem 9

Determine if the functions (x^2)+x-2, 2(x^2)-3x+1, and (x^2)-6x+5 are a basis of (Subscript[P, 2])