Question #285336

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=[3113]\begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}, the matrix P that is orthogonally diagonalizes A is of the following matrices are diagonalisable

(i)P= [1/21/21/21/2]\begin{bmatrix} 1/√2 & 1/√2 \\ 1/√2 & - 1/√2 \end{bmatrix}

(ii)P= [0110]\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

(iii)P=[1/21/21/21/2]\begin{bmatrix} -1/√2 & 1/√2 \\ - 1/√2 & 1/√2 \end{bmatrix}


1
Expert's answer
2022-01-11T17:06:00-0500

18.

<u,v>=3u1v1+2u2v2<u, v> =3u_ 1 v _ 1 + 2u _ 2 v _ 2

<u,kv>=332223=1812=6<u, kv>=3\cdot3\cdot2-2\cdot2\cdot3=18-12=6

Answer: (ii) 6


19.

u±v2=u2±2<u,v>+v2||u \pm v|||^2=||u||^2\pm 2 <u, v>+||v||^2

then:

u+v2+uv2=2(u2+v2)=9+16=25||u + v|||^2+||u - v|||^2=2(||u||^2+||v||^2)=9+16=25

v=25/24=17/2||v||=\sqrt{25/2-4}=\sqrt{17/2}

Answer: (iii) Does not exist


20.

3λ113λ=0\begin{vmatrix} 3-\lambda & 1 \\ 1 & 3-\lambda \end{vmatrix}=0


λ26λ+8=0\lambda^2-6\lambda+8=0

λ1=2,λ2=4\lambda_1=2,\lambda_2=4


x+y=0    x=yx+y=0\implies x=-y

first eigenvector:

(x1x1)\begin{pmatrix} x_1 \\ -x_1 \end{pmatrix}


x+y=0    x=y-x+y=0\implies x=y

second eigenvector:

(x2x2)\begin{pmatrix} x_2 \\ x_2 \end{pmatrix}


P=(x2x1x2x1)P=\begin{pmatrix} x_2 &x_1\\ x_2&-x_1 \end{pmatrix}


Answer:

(i)P= [1/21/21/21/2]\begin{bmatrix} 1/√2 & 1/√2 \\ 1/√2 & - 1/√2 \end{bmatrix}

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