column vectors

Examples of column vectors in the following topics:

• 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.

• Cramer's Rule

  • It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.

• 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.  
  • Take the first row of Matrix $A$ and multiply by the first column of Matrix $B$:  The first element of $A$ times the first column element of $B$, plus the second element of $A$ times the second column element of $B$.

• Cofactors, Minors, and Further Determinants

  • 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 $j$.
  • We will find the determinant of the following matrix A by calculating the determinants of its cofactors for the third, rightmost column and then multiplying them by the elements of that column.

• Application of Systems of Inequalities: Linear Programming

  • First, a nonzero pivot element is selected in a nonbasic column.
  • The result is that if the pivot is in row $r$, then the column becomes the $r$-th column of the identity matrix.
  • The variable for this column is now basic, replacing the variable which corresponded to the $r$-th column of the identity matrix.
  • For the choice of pivot row, only positive entries in the pivot column are considered.
  • Columns 2, 3, and 4 can be selected as pivot columns; for this example column 4 is selected.

• Addition and Subtraction; Scalar Multiplication

  • In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction.

• Simplifying Matrices With Row Operations

  • Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
  • 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: $x=2, y=3, z=-1$.

• Sums and Series

  • Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group.

• Matrices and Row Operations

  • The row space of a matrix is the set of all possible linear combinations of its row vectors.

• The Remainder Theorem and Synthetic Division

  • Then add the next column of coefficients, get the result and multiply that by the divisor to find the third coefficient $-27$: