Week 2

Question 1

What is the general solution?

Answer 1

The solution to a set of linear equations where free variables are put in terms of any real number and the number of solutions is infinite.

Question 1

What does it mean when an SLE is inconsistent?

Answer 1

There are no possible solutions so the solution set of the SLE equals null. Also the matrix will likely contain a nonense equation like 0x = 1.

Question 2

What are the two possible cases when an SLE is consistent? When do they occur?

Answer 2

An infinite number of solutions.
A unique solution.
An infinite number of solutions occurs when there is at least one free variable remaining in the RREF of the augmented matrix.
A unique solution occurs when every variable’s column contains a leading one in RREF.

Question 3

Define vector space.

Answer 3

R^2 := {(x, y) : x, y elememts of R}

Question 4

How is a column vector in R^2 written?

Answer 4

(x
y)

Question 5

What is another name for R^n? Define R^n.

Answer 5

N-dimensional euclidean space.
R^n := {(X1, X2, …, Xn): Xi element of real}

Question 6

Define the addition of two vectors. Define scalar multiplication.

Answer 6

Vector + vector = vector
scalar * vector = vector

Question 7

Define the modulus of a vector.

Answer 7

The modulus of the vector (x y) in R^2 is given by ¦(x y)¦ := sqrt(x^2 + y^2)

Question 8

Define the modulus/length of a vector x element of R^n

Answer 8

¦x¦ = sqrt(X1^2 + X2^2 + … + Xn^2)

where x = (X1 X2 … Xn)

Question 8

Define the dot product or scalar product of two vectors.

Answer 8

x.y := X1Y1 + … + XnYn
where x = (X1 … Xn) and y = (Y1 … Yn)

Question 9

Define a matrix of m x n.

Answer 9

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

Question 10

Define matrices with the same size and dimension.

Answer 10

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 * n matrices for some positive integers m and n.

Question 11

Define the method of position in a matrix.

Answer 11

If A is an m * n matrix, the entry appearing in the ith row and jth column of A called the (i, j) position is denoted Aij.

A = Aij where i = 1 … m and j = 1 … n

Question 12

Define the sum of two matrices: A + B.

Answer 12

A and B are matrices of the same size (m * n ). We define the sum of A + B to be the m * n matrix whose entries are given by

(A + B)ij = (A)ij + (B)ij for i = 1 … m and j=1 … n

Question 13

Define the scalar multiplication of c and matrix Aij.

Answer 13

Let A be a m * n matrix and let c be a real number. Then cA is the m * n matrix with entries defined by

(cA)ij = c(A)ij

Question 14

What are the five laws of matrix arithmetic?

Answer 14

A + B = B + A (+ commutative)
(A + B) + C = A + (B + C) (+ associative)
AB not equal to BA (* not commutative)
(AB) * C = A * (B * C) ( associative)
Distributive laws: * over +: (A + B)C = AC + BC

Question 15

Define Av the product of the matrix A with the vector v.

Answer 15

Let A=(aij) a m * n matrix and let v = (vj) be a vector with n entries (an n * 1 matrix). Then the product Av is defined to be the m * 1 matrix.

(a11 * v1 + a12 * v2 … a1n * vn)
(a21 * v1 + a22 * v2 … a2n * vn)
. . . . . . . . . . . . . . . . . . . . . . . .
(am1 * v1 + am2 * v2 + … + amn * vn)