Linear Algebra Answers

. Find a basis for the orthogonal complement of the vector v(1, 3,−2) of the euclidean vector space (R 3 , ·). Argue whether this basis is orthonormal or not.

Study whether the vectors v1(1, 1, 2), v2(2, 3, 0), v3(3, 4, 2) in R 3 are linearly dependent. If so, find the linear dependency relation

 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