Linear Algebra Answers

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.

Q4.

The value of x "\\in" R^4

such that (4, 3, - 3, 2) + 3x = (7, -3, 3, 2) is

(1) x = (4, -4, 3, 0)

(2) x = (4, - 4, 3, 0)

(3) x = (1, -2, 0, 0)

(4) None of the given answers is true.

Q5.

Let W be a subset of R^3 defined as

W = (x, y, z) "\\in" R^3: 2x + y - z - 1 = 0.

Then

(1) W is a subspace of R^3

(2) W is closed under scalar multiplication

(3) W is not a subspace of R^3

(4) None of the given answers is true.

Q6.

The set of differentiable real-valued functions f on the interval (0,3) such that f'(2) = "\\alpha" is a subspace of R^(0,3). The value of must be

(1) negative

(2) positive

(3) zero

(4) None of the given answers is true.

Q1.

Consider the sets

A = {x "\\in" Z : x = 2 (y - 2) for some

y "\\in" Z} and B = {x "\\in" Z : x = 2z for some z "\\in" Z}. Then

(i) A and B are equal,

(ii) A {\displaystyle \subsetneq } B,

(iii) B {\displaystyle \subsetneq } A

(1) (i) only

(2) (ii) only.

(3) (iii) only

(4) None of the given answers is true.

Q2.

Consider a function f: R → R^+ defined as f(x) =e^x: Then the image of S ={x ∈ R: 0 ≤ x^2 - 9} is

(i) (-∞; e^-3]U[e^3;∞);

(ii) [e^-3; e3^];

(iii)[e^3;∞)

(4) None of the given answers is true.

Q3.

The two distinct square roots of i is

(1) √2 + √2i and -√2 - √2i

(2) 2√2 + 2√2i and - 2√2 - 2√2i

(3)√2/2 + √2/2(I) and - √2/2 - √2/2(I)

(4) None of the given answers is true.

If the owner needs one employee for every 25 cups of coffee sold per hour, determine how many employees are required for the morning shift from 6 am to 1 pm?

Find the solution of the system of linear equation by using augmented matrix.

\left. \begin{array} { l } { x _ { 1 } - 2 x _ { 2 } = 0 } \\ { 3 x _ { 1 } + 4 x _ { 2 } = - 1 } \\ { 2 x _ { 1 } - x _ { 2 } = 3 } \end{array} \right.

find the inverse of this matrix

A= 1 3 5 4

7 2 9 6