1 | review matrix, vector spaces

Question 1

m x n ?

Answer 1

m rows
n columns

Question 2

Additive identity of a matrix

Answer 2

Zero matrix of the same order is an additive identity

A + 0 = 0 + A = A

Question 3

Additive inverse?

Answer 3

−𝐴 is the additive inverse, since
𝐴 + (−𝐴) = (−𝐴) + 𝐴 = 0

Question 4

If 𝐴 is an 𝑚 × 𝑛 matrix and 𝐵 is an 𝑛 × 𝑝 matrix,

the product 𝐶 is an _? x _? matrix whose entries are given by _______?

What is this product called?

Answer 4

m x p

c_ij = a_i. b_.j

dot product

Question 5

orthogonal vectors?

example?

Answer 5

Two vectors are said to be orthogonal (perpendicular) if their dot product is zero.

Example. 𝑎 = 1 2 3 , 𝑏 = −2; 1; 0
𝑎 ∙ 𝑏 = 0

Question 6

Is matrix multiplication (dot product) commutative?

Answer 6

no

Question 7

Square matrix - definitiion?

diagonal of a square matrix?

Answer 7

A square matrix is a matrix which has the same number of rows and columns.

The diagonal of a square matrix is given by the entries which have the same row and column indices.

Question 8

Diagonal matrix?

Answer 8

A diagonal matrix is a square matrix whose all off-diagonal entries are zero

Question 9

Identity matrix?

Operations?

Answer 9

The identity matrix 𝐼_𝑛 is a diagonal matrix whose all diagonal entries are 1.

For an 𝑚 × 𝑛 matrix 𝐴, it holds that 𝐼_m𝐴 = 𝐴 and 𝐴𝐼_𝑛 = 𝐴
Examples.

Question 10

For an 𝑚 × 𝑛 matrix 𝐴, it holds that ____𝐴* = 𝐴

Answer 10

𝐼_𝑚*𝐴

Question 11

For an 𝑚 × 𝑛 matrix 𝐴, it holds that 𝐴*____ = 𝐴 ?

Answer 11

𝐴*𝐼_𝑛

Question 12

Matrix operation:

commutative law?

Answer 12

only for addition!

𝐴 + 𝐵 = 𝐵 + 𝐴

Question 13

Matrix operatiom:

Scalar multiplication law?

Answer 13

(𝑟𝐴)𝐵 = 𝐴(𝑟𝐵) = 𝑟(𝐴𝐵)

Question 14

Matrix operation:

Distributive law?

Answer 14

(𝐴 + 𝐵)𝐶 = 𝐴𝐶 + 𝐵𝐶
𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶

Question 15

Matrix operation:

Associative law?

Answer 15

(𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶)
(𝐴𝐵)𝐶 = 𝐴(𝐵𝐶)

Question 16

Transpose matrix definition

Answer 16

If 𝐴 is an 𝑚 × 𝑛 matrix,
the transpose 𝐶 = 𝐴^𝑇 is an 𝑛 × 𝑚 matrix

whose entries are given by 𝑐_𝑖𝑗 = 𝑎_𝑗𝑖

In other words, the 𝑖 row of 𝐴 becomes the 𝑖 column in 𝐴^𝑇

Question 17

Symmetric Matrix?

Answer 17

Matrix is symmetric if 𝐴^𝑇 = 𝐴

Question 18

Assuming that all products and sums below are defined, the
following hold:

(𝐴 + 𝐵)^𝑇 = ?
(𝐴𝐵)^𝑇 = ?

Answer 18

(𝐴 + 𝐵)^𝑇 = A^𝑇 + B^𝑇
(𝐴𝐵)^𝑇 = B^𝑇A^𝑇

Question 19

Homogeneous system of linear:

Definition?

How can it be solved?

Answer 19

𝐴𝑥 = 𝑏

If all entries in 𝑏 are zero, the system is homogeneous

Otherwise, non-homogeneous

can be solved by Gaussian elimination

Question 20

Row echelon form

Definition?

Answer 20

A matrix is said to be in a row echelon form if

(i) The first non-zero entry in every row is to the right of the first non-zero entry in all the rows above

(ii) Every entry above a first non-zero entry is zero

Question 21

Row echelon form:

pivot?

Answer 21

Pivot (corner entry) = the first non-zero entry in a row

Question 23

Row Echelon Form:

reduced?

Answer 23

Reduced row echelon form = every pivot equals one!

The reduced form of matrix 𝐴 is denoted by 𝐴_𝑟𝑒𝑑

Question 24

Elementary row operations on a matrix?

Use?

Answer 24

Three elementary row operations on a matrix 𝐴 are defined as follows:

1. Interchange two rows of 𝐴
2. Multiply a row of 𝐴 by a non-zero scalar (i.e. constant)
3. Replace a row of 𝐴 by itself plus a multiple of a different row

Every matrix can be put into a (unique) reduced row echelon form by applying a (not unique) sequence of elementary row operations.

Question 25

Matrix rank

Answer 25

The reduced form of matrix 𝐴 is denoted by 𝐴_𝑟𝑒𝑑

The row rank, or simply, the rank, of a matrix 𝐴 is the number of non-zero rows in 𝐴_𝑟𝑒𝑑

Question 26

Row rank of matrix 𝐴?

𝐴 =
1 2 3
4 5 6
7 8 9
→
𝐴_𝑟𝑒𝑑 =
1 0 −1
0 1 2
0 0 0

Answer 26

2, since there are 2 non-zero rows.

Question 27

Solving homogeneous linear system:

Do elementary row operations change solution of system of equations 𝐴𝑥 = 𝑏?

Answer 27

Appication of elementary row operations does not change the solutions

Question 28

Solving homogeneous linear system:

Null space?

Answer 28

Homogeneous case.
The solution set of a homogeneous linear system 𝐴𝑥 = 0 is called the null space of 𝐴, denoted by 𝑁(𝐴) .

Reduced row echelon form can be used to determine and write down the null space of a matrix 𝐴.

Question 29

Solving homogeneous linear system:

Solution set = Null space - How to get it ?

Answer 29

Reduced row echelon form can be used to determine and write down the null space of a matrix 𝐴.

The pivot variables have non-zero coefficients, so they can be expressed in terms of the remaining variables – called, free

Question 29

Solving homogeneous linear system - example:

𝐴 =
0 1 2 0 0 0
0 3 −1 1 2 0

The matrix is already in reduced row echelon form. The pivot variables have non-zero coefficients, so they can be expressed in terms of the remaining variables – called, free.

> > show this, and show the general solution vector

Answer 29

𝑥₂ = −2𝑥₃ − 3𝑥₅ + 𝑥₆
𝑥₄ = −2𝑥₅

general solution vector is :
𝑥₁, −2𝑥₃ − 3𝑥₅ + 𝑥₆, 𝑥₃, −2𝑥₅, 𝑥₅, 𝑥₆)

Question 30

Solving homogeneous linear system - example:

𝐴 =
0 1 2 0 0 0
0 3 −1 1 2 0

…

general solution vector is :
𝑥₁, −2𝑥₃ − 3𝑥₅ + 𝑥₆, 𝑥₃, −2𝑥₅, 𝑥₅, 𝑥₆)

Write this in matrix form

Answer 30

𝑥 =
1 0 0 0
0 −2 −3 1
0 1 0 0
0 0 −2 0
0 0 1 0
0 0 0 1

Question 31

Solving homogeneous linear system

facts about num. of pivots?

Answer 31

num. of pivots + num. of free variables = num. of variables

num. of pivots = num. of constraints on the variables.

Question 32

Solving homogeneous linear system

When does 𝐴𝑥 = 0 have a unique solution?

Answer 32

Note that the zero-vector 0 is always a solution: trivial solution If the solution is unique, then 𝑁 𝐴 = {𝟎} (This happens when there are no free variables)

otherwise, if there is a free variable, there must be non trivial solution

Therefore, a homogeneous system has a unique solution if and only if every variable is a pivot variable!

Question 33

Solving non-homogeneous linear system

Answer 33

Appication of elementary row operations does not change the solutions to the system of equations 𝐴𝑥 = 𝑏; this is applied to the augmented matrix 𝐴|𝑏

Non-homogeneous case.
If a system with the augmented matrix has a particular solution 𝑝, then any other solution has the form 𝑝 + 𝑥 where

𝑥 ∈ 𝑁(𝐴).

Question 34

System of linear equations is ‘consistent’?

Answer 34

either one solution or infinitely many solutions.

Question 35

System of linear equations is ‘inconsistent’?

Answer 35

A system of linear equations is said to be inconsistent if it has no solution

Question 36

Solving homogeneous linear system

what are free variables?

Answer 36

The non-pivot, non-zero variables in a matrix that is in reduced echelon form

Question 37

Formula for number of free variables?

Answer 37

num. free var = num. var - num. pivots
= num. var - num. constraints