Chapter 5, Problem 1E

Set $x=\left[0:4,4,-1,1\right]\text{'}$ and $y=\text{ones}\left(9,1\right)$

1. Use the MATLAB function norm to compute the values of $\Vert x\Vert ,\Vert y\Vert ,\Vert x+y\Vert$ and to verify that the triangle inequality holds. Use MATLAB also to verify that the parallelogram law ${\Vert x+y\Vert }^2+{\Vert x-y\Vert }^2=2\left({\Vert x\Vert }^2+{\Vert y\Vert }^2\right)$ is satisfied.
2. If $t=\frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }$ then why do we know that $\left|t\right|$ must be less than or equal to 1? Use MATLAB to compute the value of t and use the MATLAB function acos to compute the angle between x and y. Convert the angle to degrees by multiplying by $180/\pi .$ (Note that the number p is given by pi in MATLAB.)
3. Use MATLAB to compute the vector projection pofxontoy.Set $z=x-p$ andverifythat z is orthogonal to p by computing the scalar product of the two vectors. Compute ${\Vert x\Vert }^2$ and ${\Vert z\Vert }^2+{\Vert p\Vert }^2$ and verify that the Pythagorean law is satisfied.

To determine

To Compute:the values of $\Vert x\Vert$ , $\Vert y\Vert$ , $\Vert x+y\Vert$ by using the MATLAB function and verify that the triangle inequality holds and verify the parallelogram law by using MATLAB.

Answer to Problem 1E

The parallelogram law is verified using MATLAB.

Explanation of Solution

Given:

  $x=\left[0:4,4,-4,1,1\right];$

  $y=ones\left(9,1\right)$

Calculation:

We set the matrices $x$ and $y$ and compute $\Vert x\Vert$ , $\Vert y\Vert$ and $\Vert x+y\Vert$

using MATLAB as shown below:

Input:

  $\begin{array}{l}\gg x=\left[0:4,4,-4,1,1\right];\\ \gg y=ones\left(9,1\right);\\ \gg norm\left(x\right)\end{array}$

Output:

Ans=

  $8$

Input:

  $\gg norm\left(y\right)$

Output:

Ans=

  $3$

Input:

  $\gg norm\left(x+y\right)$

Output:

Ans=

  $9.8489$

The parallelogram law is verified using MATLAB as shown below.

Input:

  ${\left(norm\left(x+y\right)\right)}^{\wedge }2+{\left(norm\left(x-y\right)\right)}^{\wedge }2$

Output:

Ans=

  $146$

Input:

  $\gg 2^{\ast }{\left(norm\left(x+y\right)\right)}^{\wedge }2+{\left(norm\left(y\right)\right)}^{\wedge }2$

Output:

Ans=

146

Conclusion:

Hence, the values of $\Vert x\Vert =8$ , $\Vert y\Vert =3$ , $\Vert x+y\Vert =9.8489$

To determine

To Compute: the value of t and acosto compute the angle between x and y

Answer to Problem 1E

The value $t=0.5000$ and the angle between $x$ and $y$ is $60.000$

Explanation of Solution

Given:

  $t=\frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }$

  $\begin{array}{l}\left|x^{T}y\right|\le \Vert x\Vert \Vert y\Vert \\ \frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }\le 1\\ \left|t\right|\le 1\end{array}$

Calculation:

Consider that $t=\frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }$

From Cauchy-Schwarz inequality, if $x$ and $y$ are vectors in either or, then

  $\begin{array}{l}\left|x^{T}y\right|\le \Vert x\Vert \Vert y\Vert \\ \frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }\le 1\\ \left|t\right|\le 1\end{array}$

[Since $t=\frac{x^{T}y}{\Vert x\Vert \Vert y\Vert }$ ]

The value of $t$ and the angle between $x$ and $y$ is computed using MATLAB as shown below.

Input:

  $\gg t=\left(x\ast y\right)/\left(norm\left(x\right)\ast norm\left(y\right)\right)$

Output:

  $t=0.5000$

Input:

  $\gg$ acos(t)

Output:

Ans=

  $1.0472$

Input:

  $\gg acos\left(t\right)\ast \left(180/pi\right)$

Output:

Ans=

  $60.000$

Conclusion:

Thus,the value $t=0.5000$ and the angle between $x$ and $y$ is $60.000$

To determine

To Compute: the vector projection p of x onto Y and verify z is orthogonal to P by computing the scalar product of the two vectors.

Answer to Problem 1E

The inner product of z, p is zero and hence both are orthogonal.

Explanation of Solution

Given:

  $z=x-p$

  $p=\frac{x^{T}y}{y^{T}y}y$

  $\alpha =\frac{x^{T}y}{\Vert y\Vert }$

Calculation:

The vector and scalar projection of x onto y is given as $p=\frac{x^{T}y}{y^{T}y}y$ and $\alpha =\frac{x^{T}y}{\Vert y\Vert }$ respectively

The MATLAB program is as shown below.

Input:

  $\gg p=\left(\left(x\ast y\right)/\left(x\ast y\right)\right)\ast y$

Output:

P=

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $1.3333$

  $\gg z=x-p$

  $\gg alpha=\left(x\ast y\right)/norm\left(y\right)$

Alpha=

  $4$

  $\gg {\left(norm\left(x\right)\right)}^{\wedge }2$

Ans=

  $64$

Input:

  $\gg {\left(norm\left(z\right)\right)}^{\wedge }2+{\left(norm\left(p\right)\right)}^{\wedge }2$

Output:

Ans=

  $64.0000$

Hence, the Pythagorean law is satisfied. Therefore the vector z,p is orthogonal.

Also we can verify using MATLAB command as follows:

Input:

  $\gg z\ast p$

Output:

Ans=

  $0$

Therefore the inner product of z, p is zero and hence both are orthogonal.

Conclusion:

Hence, the inner product of z, p is zero and hence both are orthogonal.