Defenitions

Question 1

Define leading variable.

Answer 1

The variables corresponding to columns containing leading 1’s.

Question 2

Define free variable.

Answer 2

A variable whose column in RREF does not contain a leading 1.

Question 3

Define R^2 vector.

Answer 3

A vector of length 2.
R^2 := {(x, y) : x , y elements of R}

Question 4

Define R^n vector.

Answer 4

A vector of length n.
R^n := {(x1….xn) : xi elements of R}

Question 5

Define the length or the modulus of a vector (x y) in R^2.

Answer 5

|(x y)| := sqrt(x^2 + y^2)

Question 6

Define the length or the modulus of a vector x elements of R^n.

Answer 6

|x| := sqrt(x1 ^2 + … + xn ^2)
where x = (x1 … xn)

Question 7

Define the scalar product or dot product of two vectors x and y.

Answer 7

<x, y> := x1y1 + … + xnyn
where x = (x1 .. xn) and y = (y1 … yn)

Question 8

Define a matrix.

Answer 8

An m x n real matrix is an array of mn real numbers positioned in m rows and n columns.

Question 9

Define matrices of the same size.

Answer 9

Two matrices are said to have the same size or dimension if they have the same number of rows and the same number of columns, that is if they are both m x n matrices for some positive integers m and n.

Question 10

Define the transpose of A.

Answer 10

Let A = (a i j), m x n matrix. A^T is the n x m matrix A^T := a j i

Question 11

Define symmetric matrix.

Answer 11

A n x n matrix A is called symmetric if A = A^T

Question 12

Define a square matrix.

Answer 12

A matrix is called square if its size is n x n for some positive integer n. The set of n x n matrices with real entries is denoted Mn(R).

Question 13

Define the identity matrix.

Answer 13

The n x n identity matrix In is the matrix that has on the diagonal the entries 1 and 0 elsewhere.

Question 14

Define the minor of A.

Answer 14

For each (i, j) entry of a n x n square matrix A, we define the minor Mij of A to be the determinant of the (n - 1) x (n - 1) submatrix of A found by deleting the ith row and jth column of A.

Question 15

Define the matrix of minors.

Answer 15

The matrix of minors of an n x n matrix A is the n x n matrix M whose (i, j) entry is the (i, j) minor of A.

Question 16

Define the matrix of cofactors.

Answer 16

The matrix of cofactors of an n x n matrix A is the n x n matrix C whose (i, j) entry is (-1)^i + j(Mij), where Mij is the (i, j) minor of A.

Question 17

Define the adjoint of the matrix A.

Answer 17

The adjoint of an n x n matrix A is the n x n matrix adj(A) that is the transpose of the matrix of cofactors of A.

Question 18

Define an elementary matrix.

Answer 18

Is a square matrix that can be obtained from the identity matrix I by performing a single elementary row operation.

Question 19

Define a homogenous system.

Answer 19

Let A be an m x n matrix and b a vector of length n. The system of linear equations Ax = b is called homogenous if the vector b of constants is the zero vector.

Question 20

Define row equivalence.

Answer 20

Let A, B be m x n matrices, we say that A and B are row-equivalent if one can be obtained from the other by a sequence of ERO’s.

Question 21

Define the rank of a matrix A.

Answer 21

Let A be an m x n matrix. Let B be the unique RREF matrix that is row-equivalent to A. The rank of A, denoted by rank(A), is number of leading ones of B.