Why choose us?

We understand the dilemma that you are currently in of whether or not to place your trust on us. Allow us to show you how we can offer you the best and cheap essay writing service and essay review service.

Linear Algebra

Linear Algebra

Chapter 3

Question 1

If the above data points actually lay on a straight line y = C+Dt, we have:


Call the matrix A and the vector on the right-hand side. Of course this system is inconsistent, but we want to find  such that  is as close as possible to . As we’ve seen, the correct choice of  is given by:

To compute this, first note that


And so

Therefore, the best-fit line for the data is:

Here are the data points and the best-fit line on the same graph:


Question 13

Let  Then, we set  and

Question 14

$\displaystyle {\mathbf v}_1 = \frac{1}{\sqrt{2}}(1,0,1,0), \; {\mathbf v}_2 = \frac{1}{\sqrt{2}}(0,1,0,1), \; {\mathbf v}_3 = \frac{1}{\sqrt{2}}(0,-1,0,1).$

We now compute $ {\mathbf w}_4.$If we denote $ {\mathbf u}_4 = (2,1,1,1)^t$then by the Gram-Schmidt process,

$\displaystyle {\mathbf w}_4$$\displaystyle =$$\displaystyle {\mathbf u}_4 - \langle {\mathbf u}_4, {\mathbf v}_1\rangle {\mat...
{\mathbf v}_2 - \langle {\mathbf u}_4, {\mathbf v}_3 \rangle {\mathbf v}_3$ 
 $\displaystyle =$$\displaystyle \frac{1}{2}(1,0,-1,0)^t.$
$\displaystyle Q = \bigl[{\mathbf v}_1, {\mathbf v}_2, {\mathbf v}_3, {\mathbf v...
...-1}{\sqrt{2}} \\
0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0
$\displaystyle R = \begin{bmatrix}\sqrt{2} & 0 & \sqrt{2} & \frac{3}{\sqrt{2}} \...
\\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & \frac{-1}{\sqrt{2}} \end{bmatrix}.$

Question 16

Chapter 4

Question 5


Question 6

Let $ \{{\mathbf u}_1, {\mathbf u}_2, \ldots, {\mathbf u}_k\}$be any basis of a $ k$-dimensional subspace $ W$of $ {\mathbb{R}}^n.$Then by Gram-Schmidt orthogonalisation process, we get an orthonormal set $ \{{\mathbf v}_1, {\mathbf v}_2, \ldots, {\mathbf v}_k \} \subset {\mathbb{R}}^n$with $ W = L ( {\mathbf v}_1, {\mathbf v}_2, \ldots, {\mathbf v}_k),$and for $ 1 \leq i \leq k,$

$\displaystyle L ( {\mathbf v}_1, {\mathbf v}_2, \ldots, {\mathbf v}_i)= L ( {\mathbf u}_1, {\mathbf u}_2, \ldots, {\mathbf u}_i).$

Question 7


Question 16


Question 17

Question 24     

Question 25

Suppose we have matrix $ A = [{\mathbf x}_1, {\mathbf x}_2, \ldots, {\mathbf x}_k]$of dimension $ n \times k$with $ {\mbox{rank }} (A) = r.$Then by the application of the Gram-Schmidt orthogonalisation process yields a set $ \{{\mathbf u}_1, {\mathbf u}_2, \ldots, {\mathbf u}_r\}$of orthonormal vectors of $ {\mathbb{R}}^n.$In this case, for each $ i, \; 1 \leq i \leq r,$we have

$\displaystyle L({\mathbf u}_1, {\mathbf u}_2, \ldots, {\mathbf u}_i) = L({\math...
\ldots, {\mathbf x}_j), \; {\mbox{ for some }} \; j, \;\; i \leq j \leq k.$

Question 14

Question 3



Bretscher, O. (2004). Linear Algebra with Applications, (3rd ed.).  New York, NY: Prentice Hall.

Farin, G., & Hansford, D. (2004). Practical Linear Algebra: A Geometry Toolbox. London: AK Peters.

Friedberg, S. H., Insel, A. J., & Spence, L. E. (2002). Linear Algebra, (4th ed.). New York, NY: Prentice Hall.

Leon, S. J. (2006). Linear Algebra with Applications, (7th ed.). New York, NY: Pearson Prentice Hall.

McMahon, D. (2005). Linear Algebra Demystified. New York, NY: McGraw–Hill Professional.

Zhang, F. (2009). Linear Algebra: Challenging Problems for Students. Baltimore, MA: The Johns Hopkins University Press.

All Rights Reserved, scholarpapers.com
Disclaimer: You will use the product (paper) for legal purposes only and you are not authorized to plagiarize. In addition, neither our website nor any of its affiliates and/or partners shall be liable for any unethical, inappropriate, illegal, or otherwise wrongful use of the Products and/or other written material received from the Website. This includes plagiarism, lawsuits, poor grading, expulsion, academic probation, loss of scholarships / awards / grants/ prizes / titles / positions, failure, suspension, or any other disciplinary or legal actions. Purchasers of Products from the Website are solely responsible for any and all disciplinary actions arising from the improper, unethical, and/or illegal use of such Products.