 ## There is a board with 2 rows and n columns represented by a matrix

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. A simple question: By definition, does an m x n matrix have m rows and n columns, or is it vice versa? I suggest you always to check the notation on the book which you are using.

I found sometimes this notation with different meaning. In advanced books, for example. Even the notation for linear maps as matrices. Always check and make sure you have the right convention for the occasion. Usually m x n is rows x columns.

However, when addressing elements within a matrix, it's the opposite. The element "a sub i,j" references the element in the ith row and jth column. Always check to make sure you have the correct convention! It's m-rows and n-Columns. Here is an example, how you can generate and read a matrix in JavaScript :. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How many rows and columns are in an m x n matrix? Ask Question. Asked 7 years, 10 months ago. Active 1 year, 4 months ago. Viewed k times. Anderson Green Anderson Green 7 7 gold badges 16 16 silver badges 23 23 bronze badges. Curiously "m by n matrix" is about twice as common as "n by m matrix" in Google search results.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Given a NxN matrix with 0s and 1s. Set every row that contains a 0 to all 0 s and set every column that contains a 0 to all 0 s.

A Microsoft Engineer told me that there is a solution that involves no extra memory, just two boolean variables and one pass, so I'm looking for that answer. Ok, so I'm tired as it's 3AM here, but I have a first try inplace with exactly 2 passes on each number in the matrix, so in O NxN and it is linear in the size of the matrix. So the first pass sets the markers and resets the rest to 0's.

I doubt strongly that I can be done in 1 pass as squares in the beginning depend on squares in the end. Maybe my 2nd pass can be made more efficient This cannot be done in one pass since a single bit has an effect on bits before and after it in any ordering. IOW Whatever order you traverse the array in, you may later come accross a 0 which means you have to go back and change a previous 1 to a 0.

People seem to think that by restricting N to some fixed value say 8 you can solve this is one pass. Well that's a missing the point and b not the original question. I wouldn't post a question on sorting and expect an answer which started "assuming you only want to sort 8 things That said, it's a reasonable approach if you know that N is in fact restricted to 8. My answer above answers the original question which has no such retriction. In any case set the current element to 0.

Do a linear scan through the final row and column and looking for 1s. Set 1s in the corresponding elements in body of the matrix where the final row and column are both 1s. I've got a solution here, it runs in a single pass, and does all processing "in place" with no extra memory save for growing the stack. It uses recursion to delay the writing of zeros which of course would destroy the matrix for the other rows and cols:. I don't think it's doable. When you're on the first square and its value is 1, you have no way of knowing what the values of the other squares in the same row and column are. So you have to check those and if there's a zero, return to the first square and change its value to zero. I'll recommend doing it in two passes - the first pass gathers information about which rows and columns must be zeroed out the information is stored in an array, so we're using some extra memory.

The second pass changes the values. I know that's not the solution you're looking for, but I think it's a practical one. The constraints given by you render the problem unsolvable. Another solution that takes two passes, is to accumulate ANDs horizontally and vertically:. I thought I could design such an algorithm using parity bitsHamming codes or dynamic programmingpossibly using those two booleans as a 2-bit number, but I've had no success yet.

Can you please re-check the problem statement with your engineer and let us know? If there is indeed a solution, I want to keep chipping away at the problem.

That should do it, in a single pass -- but there is an assumption here that N is small enough for the CPU to do arithmetic on a single row, else you're going to need to loop over each row to determine if it's all 1s or not, I believe.Throughout, boldface is used for the row and column vectors. The transpose indicated by T of a row vector is a column vector. The set of all row vectors forms a vector space called row spacesimilarly the set of all column vectors forms a vector space called column space.

The dimensions of the row and column spaces equals the number of entries in the row or column vector. The column space can be viewed as the dual space to the row space, since any linear functional on the space of column vectors can be represented uniquely as an inner product with a specific row vector. To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons see alternative notation 2 in the table below.

Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix. The dot product of two vectors a and b is equivalent to the matrix product of the row vector representation of a and the column vector representation of b. The matrix product of a column and a row vector gives the outer product of two vectors a and ban example of the more general tensor product.

The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product. Continuing with row vectors, matrix transformations further reconfiguring n -space can be applied to the right of previous outputs.

The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation. Nevertheless, using the transpose operation these differences between inputs of a row or column nature are resolved by an antihomomorphism between the groups arising on the two sides.

The technical construction uses the dual space associated with a vector space to develop the transpose of a linear map. For an instance where this row vector input convention has been used to good effect see Raiz Usmani,  where on page the convention allows the statement "The product mapping ST of U into W [is given] by:.

Ludwik Silberstein used row vectors for spacetime events; he applied Lorentz transformation matrices on the right in his Theory of Relativity in see page In when McGraw-Hill published Differential Geometry by Heinrich Guggenheimer of the University of Minnesotahe used the row vector convention in chapter 5, "Introduction to transformation groups" eqs.

When H. Coxeter reviewed  Linear Geometry by Rafael Artzyhe wrote, "[Artzy] is to be congratulated on his choice of the 'left-to-right' convention, which enables him to regard a point as a row matrix instead of the clumsy column that many authors prefer.

Hirschfeld used right multiplication of row vectors by matrices in his description of projectivities on the Galois geometry PG 1,q. In the study of stochastic processes with a stochastic matrixit is conventional to use a row vector as the stochastic vector.

From Wikipedia, the free encyclopedia. Van Nostrand Company.In linear algebrathe column space also called the range or image of a matrix A is the span set of all possible linear combinations of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.

This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces R n and R m respectively. If one considers the matrix as a linear transformation from R n to R mthen the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. The concept of row space generalizes to matrices over Cthe field of complex numbersor over any field. Intuitively, given a matrix Athe action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients.

Another way to look at this is that it will 1 first project x into the row space of A2 perform an invertible transformation, and 3 place the resulting vector y in the column space of A. See singular value decomposition for more details on this second interpretation. Since these four row vectors are linearly independentthe row space is 4-dimensional.

Let K be a field of scalars. A linear combination of these vectors is any vector of the form. The set of all possible linear combinations of v 1That is, the column space of A is the span of the vectors v 1Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:. This is the same as the image or range of the corresponding matrix transformation. The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space.

The columns of this matrix span the column space, but they may not be linearly independentin which case some subset of them will form a basis. To find this basis, we reduce A to reduced row echelon form :. At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two. Therefore, the first, second, and fourth columns of the original matrix are a basis for the column space:.

Note that the independent columns of the reduced row echelon form are precisely the columns with pivots. This makes it possible to determine which columns are linearly independent by reducing only to echelon form. The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set.

To find the basis in a practical setting e. The dimension of the column space is called the rank of the matrix. The rank is equal to the number of pivots in the reduced row echelon formand is the maximum number of linearly independent columns that can be chosen from the matrix.

Because the column space is the image of the corresponding matrix transformationthe rank of a matrix is the same as the dimension of the image.All rights reserved. If you see any errors or have comments, please let us know. An array keeps track of multiple pieces of information in linear order, a one-dimensional list. However, the data associated with certain systems a digital image, a board game, etc.

## Describing Matrices (Rows and Columns)

To visualize this data, we need a multi-dimensional data structure, that is, a multi-dimensional array. A two-dimensional array is really nothing more than an array of arrays a three-dimensional array is an array of arrays of arrays. Think of your dinner. You could have a one-dimensional list of everything you eat:. Or you could have a two-dimensional list of three courses, each containing two things you eat:. In the case of an array, our old-fashioned one-dimensional array looks like this:.

For our purposes, it is better to think of the two-dimensional array as a matrix. A matrix can be thought of as a grid of numbers, arranged in rows and columns, kind of like a bingo board.

We might write the two-dimensional array out as follows to illustrate this point:. We can use this type of data structure to encode information about an image. For example, the following grayscale image could be represented by the following array:. To walk through every element of a one-dimensional array, we use a for loop, that is:. For a two-dimensional array, in order to reference every element, we must use two nested loops.

This gives us a counter variable for every column and every row in the matrix. For example, we might write a program using a two-dimensional array to draw a grayscale image. Each cell is a rectangle whose brightness oscillates from with a sine function.

Processing p5. Two-dimensional Arrays Daniel Shiffman An array keeps track of multiple pieces of information in linear order, a one-dimensional list. Processing was initiated by Ben Fry and Casey Reas.

It is developed by a small team of volunteers.What is row and column matrix? Matrices with a single row are called row vectors, and those with a single column are called column vectors. See Full Answer. What is the difference from a row and a column?

Alternatively, in a table which i assume you're referring torows are 'horizontal' collections of items belonging to a certain category; Columns are 'vertical' collections. Each item orange in the table 'belongs' to one row yellow and one column red well whats the difference between them. Row and column vectors. The terms vertical and horizontal often describe directions: a vertical line goes up and down, and a horizontal line goes across.

You can remember which direction is vertical by the letter, "v," which points down. In a relational database, a column is a set of data values of a particular simple type, one value for each row of the database. A column may contain text values, numbers, or even pointers to files in the operating system. A column can also be called an attribute. What is a row and column in a table? In the context of a relational database, a row —also called a tuple—represents a single, implicitly structured data item in a table.

In simple terms, a database table can be thought of as consisting of rows and columns. Each column expects a data value of a particular type.

What comes first row or column? This is read aloud, "two by three. Are vectors row or column?

Linear Algebra 14TBD: The Direct Algebraic Definition of the Determinant

Is a 1x1 matrix the same as a scalar?When we need to read out the elements of an array, we read it out row by row. In a matrix A, an element in row i and column j is represented by a ij.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. In these lessons, we will learn what is a matrix? The following diagram shows the rows and columns of a 3 by 2 matrix. Scroll down the page for more examples and solutions.

The dimensions or order of a matrix gives the number of rows followed by the number of columns in a matrix. We usually denote a matrix by a capital letter.

Each element is defined by its position in the matrix. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. ### Nilar  