Linear Algebra Answers

Define a vector subspace of a vector space V

Define the kernel and the image of a linear transformation K : M → N and hence Show that the Kernel and image of K are vector subspaces of M and N respectively

Find a 2×2 matrix A such that A^2 is a diagonal but not A

18. Suppose R^2 has weighted inner product given as <u, v> =(3u base 1 v base 1 + 2u base 2 V base 2 for u = (u base 1, u base 2), v = (V base 1, v base 2). Let u = (1, 2), v = ( 2, - 1) and K = 3. Then the valued of <u, kv> is.....

(i) 4

(ii) 6

(iii) 18

(iv) None

19. Suppose that u, v "\\isin" V are such that ||u|| = 2, ||u +v|| = 3 and ||u - v|| = 4. Then ||v|| is?

(i) 17/2

(ii) √17

(iii) Does not exist

(iv) None

20. For a given matrix A="\\begin{bmatrix}\n3 & 1 \\\\\n 1 & 3\n\\end{bmatrix}", the matrix P that is orthogonally diagonalizes A is of the following matrices are diagonalisable

(i)P= "\\begin{bmatrix}\n1\/\u221a2 & 1\/\u221a2 \\\\\n 1\/\u221a2 & - 1\/\u221a2\n\\end{bmatrix}"

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

(iii)P="\\begin{bmatrix}\n-1\/\u221a2 & 1\/\u221a2 \\\\\n - 1\/\u221a2 & 1\/\u221a2\n\\end{bmatrix}"

15. Suppose T : R^2"\\to" R^3 is linear defined by T(x, y) =(x + 3y, x - y, x). Then

(i) 1

(ii) 2

(iii) 3

(iv) None

16. Suppose T : R^3"\\to" R^3 is linear and has an upper-triangular matrix with respect to the basis (1,0,0),(1,1,1),(1,1,2). Then, the orthonormal basis of R^3 with respect to which T has an upper-triangular matrix is...

(i) (1, 0, 0), (0, 1/(√2), 1/(√2)), (0, - 1/(√2), 1/(√2))

(ii) (1, 0, 0), (0, 1, 0), (0, 1/(√2), - 1/(√2)

(iii) (1, 0, 0), (0, - 1, 1), (0, 1, 1)

(iv) None

17. Which of the following defines an inner product

(i) <(x base 1, x base 2), y base 1, y base 2)>2x base 1 y base 1 +x base 2 y base 2 in R^2

(ii) <(x base 1, x base 2), y base 1, y base 2)>x base 1 y base 1 +2x base 2 y base 2 - 1 in R^2

(iii) <a base 1 + b base 1 x +c base 1 x^2, a base 2 +b base 2 x + c base 2 x^2 > = a base 1 b base 1 +a base 2 b base 2 +c base 1 c base 2 in P base 2

13. 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(TS) =

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

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

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

(iv) None

14. Suppose T : R^2"\\to" R^2 is linear defined by T(x, y) = (y, x). Then the eigenvalues of T is...

(i) 1 and - 1

(ii) 0 and 2

(iii) Does not exist

(iv) None

8. 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) "\\isin" R^5: x base 1 = 3x base 2 and x base 3 =7x base 4}. Then the basis of W is

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

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

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

9. The basis of a solution space of given homogeneous 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 =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

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

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

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

(iv) None

10. 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

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

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

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

(iv) None

Find wheather the vectors (2 0 0 0) (2 0 0 8) (0 0 0 0) and (2 0 9) are linearly dependent or not

(i) Using Cayley Hamilton theorem, find 𝐴

8 − 𝐴

7 + 5𝐴

6 − 𝐴

5 + 𝐴

4 − 𝐴

3 + 6𝐴

2 +

𝐴 − 2𝐼 𝑖𝑓 [

1 2 −2

2 5 −4

3 7 −5

]

1. which sets are a basis for the null space of [(1,1,-1,1),(2,1,1,4),(1,0,0,1)].
2. let T:R^3 to R^3 be defined as T(x,y,z)={x+y, x-y,x+2z). then the basis of range T is...?
3. let R^3 to R^3 defined as T(x,y,z)=(2x,x+y,x-z). then the adjoint operator T*(u,v,w) is (i) (2u+v+w,v,-w), (ii) (2u,v+w,u-w), (iii) (u,v,-w)