Linear Algebra Answers

 Study whether the vectors v1(1, 1, 2), v2(2, 3, 0), v3(0, 1, 2) in R 3 are linearly independent.

Write v as a linear combination of u1, u2, u3, where

(a) v = (4, −9, 2), u1 = (1, 2, −1), u2 = (1, 4, 2), u3 = (1, −3, 2);

(b) v = (1, 3, 2), u1 = (1, 2, 1), u2 = (2, 6, 5), u3 = (1, 7, 8);

(c) v = (1, 4, 6), u1 = (1, 1, 2), u2 = (2, 3, 5), u3 = (3, 5, 8);

Find A if (A^-1-3I)^T= 2 [-1 2

5 4]

find the minimal polynomial of the linear operator t : R³ "- R³" define by t (x,y,z) =(x+2y+3z, 4y+5z,6 z).is t

find the inverse of the following matrix A and rank of a 2. 1. 2

1. 3. 0

-1. 1. 2

2x square + y square+z square +4 yz +2 xy-2 xx

find the canonical form of the quadratic form 2x₁² +x₂² + x₃² +2x₁x₂ – 2x₁x₃ – 4x₂x₃

Q13.

Suppose T, S : R^2 "\\to" R^2 are linear defined by T (u, v) = (3u + v, u + 2v) and S (x, y) = (2x - y, x + y).

Also the matrices of T and S with respect to the standard bases of R^2 and R^2 are given as M (T) ="\\begin{bmatrix}\n 3 & 1 \\\\\n 1 & 2\n\\end{bmatrix}"

and M (S) ="\\begin{bmatrix}\n 2 & - 1 \\\\\n 1 & 1\n\\end{bmatrix}". Then M(T S) =

(1) "\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{bmatrix}"

(2) "\\begin{bmatrix}\n 3 & 1 \\\\\n 1 & 5\n\\end{bmatrix}"

(3) "\\begin{bmatrix}\n 3 & 1 \\\\\n 4 & 1\n\\end{bmatrix}"

4) None of the given answers is true

Q10.

For a given matrix A = "\\begin{bmatrix}\n 1 & 0 & 2 & -3 \\\\\n 2 & 0 & 4 & - 6 \\\\\n -3 & 0 & - 6& 9\n\\end{bmatrix}". Which of the following is true

(1) rank (A) = 3, nullity (A) = 1

(2) rank (A) = 2, nullity (A) = 2

(3) rank (A) = 1, nullity (A) = 3

(4) None of the given answers is true.

Q11.

For a, b "\\in" R, the transformation T : R^2 "\\to" R^3 defined byT (x, y) = (2x - y, 3x + y + 3A, 5x - 2y + bxy) is linear if

(1) a = b = 1

(2) a = 0; b = 1

(3) a = 1; b = 0

(4) None of the given answers is true.

Q12.

Suppose T : R^3 "\\to" R^2

is a linear defined by

T (x, y) = (4x + 3y + Z, x - 2y). Then which of the following is basis of range T is

(1) (4, 1, 3), (1, 2, 0)

(2) (1, 0), (0, 1), (1, 1)

(3) (4, 1), (3, 2), (1, 0)

(4) None of the given answers is true.

Q7.

The vectors (1, 2, 0), (0, -4, 2), (1, 0, 1) are

(1) Span R^3

(2) linearly dependent

(3) linearly independent

(4) None of the given answers is true.

Q8.

Let W be the subspace of R^5 defined by

W = {(x base 1; x base 2; x base 3; x base 4; x base 5) 2 R^5 : x base 1 = 3x base 2 and x base 3 = 7x base 4}

Then the basis of W is

(1) (3,1,0,0,1), (3,1,3,0,0), (3,1,0,0,1)

(2) (3,1,0,1,1), (0,0,3,0,1), (0,0,1,3,1)

(3) (3,1,1,0,1), (0,1,1,0,3), (0,0,1,0,1)

(4) None of the given answers is true.

Q9.

The basis for a solution space of given homogenous linear system

x base 1 + x base 2 - x base 3 = 0

x base 1 + x base 2 - x base 3 = 0

x base 1 - x base 3=0

- x base 1 + x base 3=0 "\\implies"x base 2 = 0 "\\implies"x base 2 = 0

- 2x base 1 - x base 2 + 2x base 3 = 0

- 2x base 1 - x base 2 + 2x base 3= 0

- 2x base 1 + 2x base 3 = 0 is

(1) {(1, 0, 1)}

(2) {(1, 0, 1), (0, 1, 0)}

(3) f(1; 1; 1);(1; 0; 1), (2, 1, 2)}

(4) None of the given answers is true.