row space

(noun)

The set of all possible linear combinations of its row vectors.

Related Terms

Examples of row space in the following topics:

  • Matrices and Row Operations

    • Alternatively, two $m \times n$matrices are row equivalent if and only if they have the same row space.
    • The row space of a matrix is the set of all possible linear combinations of its row vectors.
    • If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system.
    • Two matrices of the same size are row equivalent if and only if the corresponding homogeneous systems have the same set of solutions, or equivalently the matrices have the same null space.
    • Row addition (pivot): Add to one row of a matrix some multiple of another row.

  • Simplifying Matrices With Row Operations

    • Using elementary operations, Gaussian elimination reduces matrices to row echelon form.
    • By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form.  
    • Use elementary row operations to reduce the matrix to reduced row echelon form:
    • Using elementary row operations to obtain reduced row echelon form ('rref' in the calculator) the solution to the system is revealed in the last column: $x=2, y=3, z=-1$.
    • Use elementary row operations to put a matrix in simplified form

  • What is a Matrix?

    • A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
    • The size of a matrix is defined by the number of rows and columns that it contains.
    • Matrices which have a single row are called row vectors, and those which have a single column are called column vectors.
    • A matrix which has the same number of rows and columns is called a square matrix.
    • For instance, $a_{2,1}$ represents the element at the second row and first column of a matrix A.

  • Matrix Multiplication

    • When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.
    • First ask: Do the number of columns in $A$ equal the number of rows in $B$?  
    • The number of columns in $A$ is $2$, and the number of rows in $B$ is also $2$, therefore a product exists.
    • Start with producing the product for the first row, first column element.  
    • Continue the pattern with the first row of $A$ by the second column of $B$, and then repeat with the second row of $A$.

  • Further Simplification of Matrices

    • Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations.
    • In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations .
    • Matrices containing zeros below each pivot are said to be in row echelon form.
    • Then, use elementary row operations to transform A into diagonal form:
    • A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination.

  • Cofactors, Minors, and Further Determinants

    • In linear algebra, a minor of a matrix $A$ is the determinant of some smaller square matrix, cut down from $A$ by removing one or more of its rows or columns.
    • Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors.
    • A $k \times k$ minor of $A$ is the determinant of a $k \times k$ matrix obtained from $A$ by deleting $m-k$ rows and $n-k$ columns.
    • The determinant is the sum of the signed minors of any row or column of the matrix scaled by the elements in that row or column.
    • Cross out the entries that lie in the corresponding row $i$ and column $j$.

  • Binomial Expansions and Pascal's Triangle

    • The rows of Pascal's triangle are numbered, starting with row $n = 0$ at the top.
    • The entries in each row are numbered from the left beginning with $k = 0$ and are usually staggered relative to the numbers in the adjacent rows.
    • On row $0$, write only the number $1$.
    • For example, each number in row one is $0 + 1 = 1$.
    • It can be observed in the triangle that row $5$ is $1, 5, 10, 10, 5, 1$.

  • Application of Systems of Inequalities: Linear Programming

    • where the first row defines the objective function and the remaining rows specify the constraints.
    • The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the others to change the other entries in the column to $0$.
    • For the choice of pivot row, only positive entries in the pivot column are considered.
    • The values of x resulting from the choice of rows 2 and 3 as pivot rows are $\frac{10}{1}=10$ and $\frac{15}{3}=5$ respectively.
    • Of these, the minimum is 5, so row 3 must be the pivot row.

  • Complex Numbers and the Binomial Theorem

    • Recall that the binomial coefficients (from the 5th row of Pascal's triangle) are $1, 5, 10, 10, 5, \text{and}\, 1.$ Using the binomial theorem directly, we have $(2+i)^5 =2^5 + 5\cdot 2^4 i + 10\cdot 2^3 i^2 + 10\cdot 2^2 i^3 + 5\cdot 2 \cdot i^4 + i^5$.  

  • The Identity Matrix

    • The reason for this is because, for two matrices to be multiplied together, the first matrix must have the same number of columns as the second has rows.