Linear Algebra Has Immense Practical Value In Our Modern World Statistics Business Science And Engineering All Make Heavy Use Of The Subject
As a simple concrete example, consider a network of one-way roads as shown in the diagram below. The letters A, B, and C label the intersections while the numbers and the letters x, y, and z represent traffic flow rates. The flow rates could be the number cars per 10 minute period, for example.
A network of one-way roads. The letters A, B, and C label the intersections while the numbers and the letters x, y, and z represent traffic flow rates.
The basic principle that governs such a network is this: for any intersection, the inflow equals the outflow.
Because of this, we can conclude that , , and .
We can rewrite these three linear equations as a linear system in the following standard form.
The method of elimination can then be used to solve for the unknown traffic flow rates and . With three equations and three unknowns, this example can be thought of as a prototype arising in three-dimensional linear algebra.
However, it will actually have infinitely many solutions, which is actually not typical of an example of this type .
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Representation Of Problems In Linear Algebra
Lets start with a simple problem. Suppose that price of 1 ball & 2 bat or 2 ball and 1 bat is 100 units. We need to find price of a ball and a bat.
Suppose the price of a bat is Rs x and the price of a ball is Rs y. Values of x and y can be anything depending on the situation i.e. x and y are variables.
Lets translate this in mathematical form
2x + y = 100 ………..
Similarly, for the second condition-
x + 2y = 100 …………..
Now, to find the prices of bat and ball, we need the values of x and y such that it satisfies both the equations. The basic problem of linear algebra is to find these values of x and y i.e. the solution of a set of linear equations.
Broadly speaking, in linear algebra data is represented in the form of linear equations. These linear equations are in turn represented in the form of matrices and vectors.
The number of variables as well as the number of equations may vary depending upon the condition, but the representation is in form of matrices and vectors.
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Terms Related To Matrix
Order of matrix If a matrix has 3 rows and 4 columns, order of the matrix is 3*4 i.e. row*column.
Square matrix The matrix in which the number of rows is equal to the number of columns.
Diagonal matrix A matrix with all the non-diagonal elements equal to 0 is called a diagonal matrix.
Upper triangular matrix Square matrix with all the elements below diagonal equal to 0.
Lower triangular matrix Square matrix with all the elements above the diagonal equal to 0.
Scalar matrix Square matrix with all the diagonal elements equal to some constant k.
Identity matrix Square matrix with all the diagonal elements equal to 1 and all the non-diagonal elements equal to 0.
Column matrix The matrix which consists of only 1 column. Sometimes, it is used to represent a vector.
Row matrix A matrix consisting only of row.
Trace It is the sum of all the diagonal elements of a square matrix.
Endomorphisms And Square Matrices
A linear endomorphism is a linear map that maps a vector space V to itself. If V has a basis of n elements, such an endomorphism is represented by a square matrix of size n.
With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including geometric transformations, coordinate changes, quadratic forms, and many other part of mathematics.
Cramer’s rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns. Cramer’s rule is useful for reasoning about the solution, but, except for n = 2 or 3, it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense, since this determinant is independent of the choice of the basis.
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Vectors In Three Dimensions
As implied by the content in the previous section, a vector in three-dimensions is an arrow sitting inside three-dimensional space. It has a length and a direction. And it doesnt matter where its base point is.
A vector in can also be represented as a column vector that is, as a matrix of real numbers.
, then the solution set above can be written as .
It is both interesting and important to note that is one particular solution to the original system. Check this!
On the other hand, does not solve the original system. Instead, it solves the corresponding homogeneous system, where the numbers on the right-hand sides are all zeros.
This is an important observation that should be double-checked as well.
We next describe the general rules for the algebra of vectors in three-dimensional linear algebra.
Vector Algebra and Geometric Interpretations in Three-Dimensional Space
are added component-wise.
Scalar multiplication is also done component-wise.
Also note that the zero vector is and that
Visualizing the Sum of Vectors in Three-Dimensional Space
Any two nonzero vectors in which are not parallel define a plane. If we make sure all vectors are based so that their tails are at the origin, the sum of those two vectors lies in the same plane. Geometrically, the sum is determined by the parallelogram law, as discussed in Section 1.2, Vectors in Two-Dimensions.
The Dot Product and Angle Between Two Vectors
Formula for the Cross Product
The Right Hand Rule
Schools Wikipedia Selection Related Subjects: Mathematics
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces , linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.
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Linear Transformations And Change Of Bases
Linear transformation: Let V and W be two vector spaces over R, the field of real numbers. A mapping T from V to W is called a linear transformation from V to W if T = T + T for every pair of vectors u and v from V and T = T, for every vector v from V and scalar from R. In other words, T preserves the vector space structure. Equivalently, T is called a linear transformation if T = T + T, for every pair of vectors u and v from V and scalars and from R.
Let V = R3, W = R2, and let T = t. T can be verified to be a linear transformation.
Matrix representation of a linear transformation: Let V and W be an n and m dimensional vector spaces over the field of real numbers, R. Also, let BV = and BW = be ordered bases of V and W, respectively. Further, let T be a linear transformation from V into W. So, Txi, 1 i n, is an element of W and hence is a linear combination of its basis vectors that is, Txi = 1iy1 + 2iy2 + + miym, 1 i n, for some real numbers ji, 1 j m, 1 i n. In other words, there are real scalars 1i, 2i, , mi, 1 i n, such that Txi = 1iy1 + 2iy2 + + miym, for every i, 1 i n. This implies that the linear transformation T is completely determined by the real numbers ij, 1 i m, 1 j n. The m × n matrix m×n is called the matrix of T relative to the ordered bases BV and BW. In other words, m×n is the matrix representation of the linear transformation T relative to the ordered bases BV and BW.
Representing Equations In Matrix Form
Let me do something exciting for you. Take help of pen and paper and try to find the value of the matrix multiplication shown below
It can be verified very easily that the expression contains our three equations. We will name our matrices as A, X and Z.
It explicitly verifies that we can write our equations together in one place as
AX = Z
Next step has to be solution methods.We will go through two methods to find the solution.
Now, we will look in detail the two methods to solve matrix equations.
Study Guides And Outlines
- Leduc, Steven A. , Linear Algebra , Cliffs Notes, ISBN978-0-8220-5331-6
- Lipschutz, Seymour Lipson, Marc , Schaum’s Outline of Linear Algebra , McGraw-Hill, ISBN978-0-07-136200-9
- Lipschutz, Seymour , 3,000 Solved Problems in Linear Algebra, McGrawâHill, ISBN978-0-07-038023-3
- McMahon, David , Linear Algebra Demystified, McGrawâHill Professional, ISBN978-0-07-146579-3
- Zhang, Fuzhen , Linear Algebra: Challenging Problems for Students, The Johns Hopkins University Press, ISBN978-0-8018-9125-0
Multilinear Algebra And Tensors
|This section may require cleanup to meet Wikipedia’s quality standards. The specific problem is: The dual space is considered above, and the section must be rewritten for given an understandable summary of this subject. Please help improve this section if you can.|
In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space, the vector space V consisting of linear maps f: V F where F is the field of scalars. Multilinear maps T: Vn F can be described via tensor products of elements of V.
If, in addition to vector addition and scalar multiplication, there is a bilinear vector product V × V V, the vector space is called an algebra for instance, associative algebras are algebras with an associate vector product .
How To Find Eigenvectors Of A Matrix
Suppose we have a matrix A and an Eigenvector x corresponding to the matrix. As explained already, after multiplication with matrix the direction of x doesnt change. Only change in magnitude is permitted. Let us write it as an equation-
Ax = cx
x = 0 .
Please note that in the term , c denotes an identity matrix of the order equal to A multiplied by a scalar c
We have two unknowns c and x and only one equation. Can you think of a trick to solve this equation?
In equation , if we put the vector x as zero vector, it makes no sense. Hence, the only choice is that is a singular matrix. And singular matrix has a property that its determinant equals to 0. We will use this property to find the value of c.
Det = 0
Once you find the determinant of the matrix and equate to 0, you will get an equation in c of the order depending upon the given matrix A. all you have to do is to find the solution of the equation. Suppose that we find solutions as c1 , c2 and so on. Put c1 in equation and find the vector x1 corresponding to c1. The vector x1 that you just found is an Eigenvector of A. Now, repeat the same procedure with c2, c3 and so on.
Code for finding EigenVectors in python
import numpy as np
#finding the Eigenvalue and Eigenvectors of arrnp.linalg.eig
Code in R for finding Eigenvalues and Eigenvectors:
-0.3948374 0.4437557 -0.74478185 -0.5497457 -0.8199420 -0.06303763 -0.7361271 0.3616296 0.66432391
Linear Algebra One To One And Onto Function
I was just wondering how I can tell if a function is onto.$\mathbf^3\to\mathbf^1$Lets say the standard transformation matrix has vectors $\$, $\$, $\$.I know that this transformation is not one to one since they vectors are not linearly independent. How can I see if the transformation is onto?
- $\begingroup$i know but what if it’s not one to one how can i test for onto$\endgroup$Feb 18 ’14 at 21:30
- $\begingroup$But there are cases in which a function is onto but not one to one$\endgroup$Feb 18 ’14 at 21:32
- 1Feb 18 ’14 at 21:40
- $\begingroup$it was supposed to be r^3 to r^1$\endgroup$
For a linear transformation $T$ from $\mathbb^n$ to $\mathbb^n$, the following are equivalent: $T$ is one-one $T$ is onto If $T=0$, then $v=0$
Generally, for a linear transformation $T: \mathbb^n \longrightarrow \mathbb^m$, the following are equivalent: $T$ is onto There is some basis $\mathcal$ of $\mathbb^m$ such that the image of $T$ contains $\mathcal$
- $\begingroup$That’s not guaranteed because a linear transformation can send everything to 0.$\endgroup$
I generally try to use the Rank-Nullity theorem whenever I can.
The Rank Nullity Theorem. Let $f:V\rightarrow W$ be a linear map between finite-dimensional vector spaces $V$ and $W$. Then $$\operatornamef+\operatornamef=\dim V$$where $\operatornamef:=\dim\operatornamef$ and $\operatornamef:=\dim\ker f$.
So, to check if a linear map $f:\mathbb R^m\rightarrow\mathbb R^n$ is onto, we need only check that $\operatornamef=m-n$.
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Study Of Complex Systems
Most physical phenomena are modeled by partial differential equations. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells. For linear systems this interaction involves linear functions. For nonlinear systems, this interaction is often approximated by linear functions. In both cases, very large matrices are generally involved. Weather forecasting is a typical example, where the whole Earth atmosphere is divided in cells of, say, 100 km of width and 100 m of height.
Vector Space Of Linear Maps
The composition of linear maps is linear: if f are linear, then so is their composition g Z . It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.
The inverse of a linear map, when defined, is again a linear map.
If :V\to W} and :V\to W} are linear, then so is their pointwise sum f +f_} , which is defined by +f_)=f_+f_} .
is an element of the ground field K )=\alpha )} , is also linear.
itself forms a vector space over K . Furthermore, in the case that V ) } , is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.
Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
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Examples In Two Dimensions
- =0& -1\\1& 0\end}}
- by an angle counterclockwise: A
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