VECTOR ALGEBRA PDF
operations on vectors, and their algebraic and geometric properties. terminal points of a vector is called the magnitude (or length) of the vector, denoted as. Many of you will know a good deal already about Vector Algebra — how to add and Pdf copies of these notes (including larger print versions), tutorial sheets. This book is meant to provide an introduction to vectors, matrices, and least squares methods, basic topics in applied linear algebra. Our goal is to give the.
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Vector Algebra. Scalars. A physical quantity which is completely described by a single real number is called a scalar. Physically, it is something which. methods of Linear Algebra and Analytical Geometry based on the vector The Linear Algebra topics include matrix operations, determinants and systems. success and importance of vector algebra derives from the interplay between geometric The direction of a vctor V is the unit vector U parallel to V: U V V.
Moreover, Cramer's rule is unstable even for 2x2 systems. Cramer's rule can occasionally be useful in theoretical applications, but the text does not discuss these nuances at all.
Many other examples like this occur in the book, but this is perhaps the most glaring one. Clarity rating: 5 The style of the book is nice and streamlined, cutting out much unnecessary fluff, while still clearly hitting the main points.
Consistency rating: 5 The textbook follows a nice, consistent build-up of definitions and notation. The colors and styles used e. Modularity rating: 5 The author did a really excellent job of separating the sections and making the as independent as possible. I also like that there seems to be modularity in terms of complexity. The exercises also seem independent.
They have been used to classify the cancer type of a given sample or classify groups of co-regulated genes [ 5 , 6 , 7 , 8 ]. Finally, microarrays have been used to characterize biological systems using comparisons of wild-type and mutant cells, with the goal of obtaining mechanistic insights [ 9 , 10 , 11 ].
Searching for concerted, dramatic changes in gene expression or searching for differential expression of a given gene has been a successful method in analyzing transcription-profiling data, especially in description or characterization [ 4 , 12 ]. Often, investigators take a manual approach to accomplishing these tasks. However, manual approaches to data analysis are sometimes impractical or cumbersome, inspiring the development of tools to accomplish the three goals described above.
Linear Algebra and its Applications
A variety of techniques such as hierarchical clustering, k-means clustering and self-organizing maps have been implemented with success, especially in classification [ 13 ]. As the number of publicly available profiles in Saccharomyces cerevisiae alone now exceeds , a great need exists to exploit this information properly to understand cell function.
At least three independent international projects have been set up to serve as database-driven repositories of genome-wide expression data [ 14 ]. A major effort is being made to systematize data storage, especially involving XML extensible markup language , to ensure interoperability of these databases and associated analysis tools.
A related need that has been less addressed is the systematization of expression data analysis. This requirement extends not only to analysis but also to pedagogy and to practical aspects of algorithm implementation.
Advanced Linear Algebra
Various studies in the literature have successfully implemented tools from vector algebra in analyzing genome-wide expression data [ 11 , 15 , 16 ]. However, a framework for the analysis of transcription profiles using vector algebra has not yet been codified.
Here we present such a framework. Common statistical measures have natural counterparts in vector algebra that have visual interpretations and are easily implemented on a computer.
Indexing the proteins database through latent semantic indexing The contact matrix is a large matrix and it is possible to take advantage of the implicit higher-order structure in the association of terms with documents in order to improve the detection of relevant documents on the basis of terms found in queries Deerwester et al.
Considering proteins as documents and contacts from contact vectors as terms, our goal is to build a retrieval system which should be able to find conserved characteristics in structures, represented by their hydrophobic interactions, and use them to classify a huge dataset of families. All things considered, from a large matrix of term-document association data, we construct a semantic space wherein intramolecular interactions and proteins that are closely associated and placed near one another.
In other words, we can plot in 2D or 3D point representatives of proteins and contacts. SVD allows the arrangement of the space to reflect the major associative patterns in the data, and ignore the smaller, less important influences.
For instance, contacts that did not actually appear in a protein, the atoms participating in its formation may still end up close to one another in the space, if it is consistent with the major patterns of association in the data.
In conclusion, position in space serves as a semantic index and retrieval can be achieved by using the contacts of a protein as a query to identify other proteins in the same space. Users retrieve the proteins in the neighborhood of the query Deerwester et al.
Defining the similarity metric for protein structure comparison A common measure of similarity is the cosine between the query Q and the document vector D which is computed by the following: Typically, the z closest documents or all documents exceeding some cosine threshold are returned to the user.
Classifying protein structures using the similarity metric Using that similarity metric, we propose a protein structural classifier which retrieves proteins which are similar to a query based on the metric.
A First Course in Linear Algebra
In other words, each protein of the database has to be compared against the whole database. Finally we get, for each protein, all those ranked by their similarity nearness to it. To determine the effectiveness of this retrieval system, we used a well-known statistical concept of Confusion Matrix and Receiver Operating Characteristic ROC curves, Swets, A confusion matrix Provost and Kohavi, contains information about actual and predicted class assignments performed by a classifier and makes it possible to evaluate the precision of classification.
This matrix gives the true-negative, true-positive, false-negative, and false-positive rates. ROC curves are another way to examine the performance of classifiers.
An ROC graph is a plot with the false-positive rate on the X-axis and the true-positive rate on the Y-axis. The false-positive rate is the number of negative instances predicted as positives divided by the number of negative instances. The true-positive rate is the number of positive instances predicted as positives divided by the number of positive instances. In the ROC space, the point 0,1 is the perfect classifier: it classifies all positive cases and negative cases correctly.
It is 0,1 because the false-positive rate is 0 none , and the true-positive rate is 1 all. The point 0,0 represents a classifier that predicts all cases to be negative, while the point 1,1 corresponds to a classifier that predicts every case to be positive.
Point 1,0 is the classifier that is incorrect for all classifications.
NCERT Solutions for Class 12 Maths Chapter 10 - Vector Algebra
In many cases, a classifier has a parameter that can be adjusted to increase true-positives at the cost of increasing false-positives or decreasing false-positives at the cost of decreasing true-positives.
Each parameter setting provides a false-positive, true-positive pair and a series of such pairs can be used to plot an ROC curve. In our algorithms, the parameter used is a threshold that we use to decide if a protein is or is not of a given family.
An ROC curve is independent of class distribution or error costs, and it encapsulates all information contained in the confusion matrix, since false-negatives are the complement of true-positives and true-negatives are the complement of false-positives.
These curves provide a visual tool for examining the tradeoff between the ability of a classifier to correctly identify positive cases and the number of negative cases that are incorrectly classified. Another interesting feature of these curves is that the area beneath them can be used as a measure of accuracy in many applications.
Another way of comparing ROC points is by using a formula that equates accuracy with the Euclidean distance from the perfect classifier, point 0,1 on the graph.
Classifier calibration methodology There are two important parameters to be adjusted in the system. The first one refers to the number of the singular values that is used to represent the data when we reduce its dimensions.
NCERT Solutions for Class 12 Maths Chapter 10
The number of singular values has to fit all the real structures in the data and try not to bring noise and redundant information to the representation. To discover this value, there is no simple rule but one way is to try all values and choose the one that best represents the data Elden, Each parameter setting provides a false-positive, true-positive pair and a series of such pairs can be used to plot a ROC curve.
We have conduced these experiments and plotted the ROC curves and found that, for myoglobins, the optimum parameter value was 17 dimensions, as shown in Figure 1. The other parameter is related to the z number mentioned before, that is, how many proteins at the top of the rank to predict as positives. Once more, this value can be adjusted to increase true-positives at the cost of increasing false-positives or decreasing false-positives at the cost of decreasing true-positives.
Given an m vs. The r columns of the orthogonal matrices D and T define the orthonormal eigenvectors associated with the r nonzero eigenvalues of AAt and AtA, respectively, Deerwester et al. Figure 2 presents a schematic of the singular value decomposition for a mvs. The matrix A can be approximated in another matrix, Ak, by modifying the three matrices that were factored above. From S matrix, the k largest singular values may be kept and the remaining smaller ones set to zero, in order to obtain the matrix Sk.
To obtain the new matrices Dk and Tk just keep the k first columns of the corresponding matrices D and V. It is important for the LSI method that the derived Ak matrix not reconstruct the original document term matrix Ak exactly Berry et al. The SVD derives the latent semantic structure model from the orthogonal matrix, S, of singular values of A. These matrices reflect a breakdown of the original relationships into linearly-independent vectors or factor values.
In some sense, the SVD can be viewed as a technique for deriving a set of uncorrelated indexing variables or factors, whereby each term and document is represented by a vector in k-space using elements of the left or right singular vectors.
See Figure 3. The choice of the value of k is a difficult job because it is related to dimension reduction. It has to be a value that fits all the real structures in the data and but small enough so that the noise and the redundant information do not fit in the new representation.
It is important to emphasize that for modeling the problem of structural classification of proteins through their intramolecular interaction we represent the matrix A as a matrix of documents by terms and not a matrix of terms by documents as in Deerwester et al.
As said previously, the documents are the proteins and the terms are contact vectors representing the intramolecular interaction in the protein.These matrices reflect a breakdown of the original relationships into linearly-independent vectors or factor values.
Table 1 shows a sample dataset. A contact map is a compact representation of the 3D conformation of a protein. ABSTRACT In this article, we describe a novel methodology to extract semantic characteristics from protein structures using linear algebra in order to compose structural signature vectors which may be used efficiently to compare and classify protein structures into fold families. We have conduced these experiments and plotted the ROC curves and found that, for myoglobins, the optimum parameter value was 17 dimensions, as shown in Figure 1.
Various studies in the literature have successfully implemented tools from vector algebra in analyzing genome-wide expression data [ 11 , 15 , 16 ]. After the SVD computation it is possible to visualize the distribution of proteins in space, see Figure 5. Therefore this study will focus on analysis of fold-change values, hereafter called a transcription or expression profile.
This matrix gives the true-negative, true-positive, false-negative, and false-positive rates.