1 | review matrix, vector spaces Flashcards

1
Q

m x n ?

A

m rows
n columns

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2
Q

Additive identity of a matrix

A

Zero matrix of the same order is an additive identity

A + 0 = 0 + A = A

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3
Q

Additive inverse?

A

βˆ’π΄ is the additive inverse, since
𝐴 + (βˆ’π΄) = (βˆ’π΄) + 𝐴 = 0

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4
Q

If 𝐴 is an π‘š Γ— 𝑛 matrix and 𝐡 is an 𝑛 Γ— 𝑝 matrix,

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

What is this product called?

A

m x p

cij = ai. b.j

dot product

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5
Q

orthogonal vectors?

example?

A

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

Example. π‘Ž = 1 2 3 , 𝑏 = βˆ’2; 1; 0
π‘Ž βˆ™ 𝑏 = 0

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6
Q

Is matrix multiplication (dot product) commutative?

A

no

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7
Q

Square matrix - definitiion?

diagonal of a square matrix?

A

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.

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8
Q

Diagonal matrix?

A

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

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9
Q

Identity matrix?

Operations?

A

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

For an π‘š Γ— 𝑛 matrix 𝐴, it holds that 𝐼m𝐴 = 𝐴 and 𝐴𝐼𝑛 = 𝐴
Examples.

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10
Q

For an π‘š Γ— 𝑛 matrix 𝐴, it holds that ____𝐴* = 𝐴

A

πΌπ‘š*𝐴

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11
Q

For an π‘š Γ— 𝑛 matrix 𝐴, it holds that 𝐴*____ = 𝐴 ?

A

𝐴*𝐼𝑛

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12
Q

Matrix operation:

commutative law?

A

only for addition!

𝐴 + 𝐡 = 𝐡 + 𝐴

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13
Q

Matrix operatiom:

Scalar multiplication law?

A

(π‘Ÿπ΄)𝐡 = 𝐴(π‘Ÿπ΅) = π‘Ÿ(𝐴𝐡)

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14
Q

Matrix operation:

Distributive law?

A

(𝐴 + 𝐡)𝐢 = 𝐴𝐢 + 𝐡𝐢
𝐴(𝐡 + 𝐢) = 𝐴𝐡 + 𝐴𝐢

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15
Q

Matrix operation:

Associative law?

A

(𝐴 + 𝐡) + 𝐢 = 𝐴 + (𝐡 + 𝐢)
(𝐴𝐡)𝐢 = 𝐴(𝐡𝐢)

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16
Q

Transpose matrix definition

A

If 𝐴 is an π‘š Γ— 𝑛 matrix,
the transpose 𝐢 = 𝐴𝑇 is an 𝑛 Γ— π‘š matrix

whose entries are given by 𝑐𝑖𝑗 = π‘Žπ‘—π‘–

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

17
Q

Symmetric Matrix?

A

Matrix is symmetric if 𝐴𝑇 = 𝐴

18
Q

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

(𝐴 + 𝐡)𝑇 = ?
(𝐴𝐡)𝑇 = ?

A

(𝐴 + 𝐡)𝑇 = A𝑇 + B𝑇
(𝐴𝐡)𝑇 = B𝑇A𝑇

19
Q

Homogeneous system of linear:

Definition?

How can it be solved?

A

𝐴π‘₯ = 𝑏

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

Otherwise, non-homogeneous

can be solved by Gaussian elimination

20
Q

Row echelon form

Definition?

A

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

21
Q

Row echelon form:

pivot?

A

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

22
Q

Row Echelon Form:

reduced?

A

Reduced row echelon form = every pivot equals one!

The reduced form of matrix 𝐴 is denoted by π΄π‘Ÿπ‘’π‘‘

23
Q

Elementary row operations on a matrix?

Use?

A

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.

24
Q

Matrix rank

A

The reduced form of matrix 𝐴 is denoted by π΄π‘Ÿπ‘’π‘‘

The row rank, or simply, the rank, of a matrix 𝐴 is the number of non-zero rows in π΄π‘Ÿπ‘’π‘‘

25
Row rank of matrix 𝐴? 𝐴 = 1 2 3 4 5 6 7 8 9 β†’ π΄π‘Ÿπ‘’π‘‘ = 1 0 βˆ’1 0 1 2 0 0 0
2, since there are 2 non-zero rows.
26
Solving homogeneous linear system: Do elementary row operations change solution of system of equations 𝐴π‘₯ = 𝑏?
Appication of elementary row operations does not change the solutions
27
Solving homogeneous linear system: Null space?
Homogeneous case. The solution set of a homogeneous linear system 𝐴π‘₯ = 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.
28
Solving homogeneous linear system: Solution set = Null space - How to get it ?
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
28
Solving homogeneous linear system - example: 𝐴 = 0 1 2 0 0 0 0 3 βˆ’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
π‘₯2 = βˆ’2π‘₯3 βˆ’ 3π‘₯5 + π‘₯6 π‘₯4 = βˆ’2π‘₯5 general solution vector is : π‘₯1, βˆ’2π‘₯3 βˆ’ 3π‘₯5 + π‘₯6, π‘₯3, βˆ’2π‘₯5, π‘₯5, π‘₯6)
29
Solving homogeneous linear system - example: 𝐴 = 0 1 2 0 0 0 0 3 βˆ’1 1 2 0 ... general solution vector is : π‘₯1, βˆ’2π‘₯3 βˆ’ 3π‘₯5 + π‘₯6, π‘₯3, βˆ’2π‘₯5, π‘₯5, π‘₯6) Write this in matrix form
π‘₯ = 1 0 0 0 0 βˆ’2 βˆ’3 1 0 1 0 0 0 0 βˆ’2 0 0 0 1 0 0 0 0 1
30
Solving homogeneous linear system facts about num. of pivots?
num. of pivots + num. of free variables = num. of variables num. of pivots = num. of constraints on the variables.
31
Solving homogeneous linear system When does 𝐴π‘₯ = 0 have a unique solution?
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!
32
Solving non-homogeneous linear system
Appication of elementary row operations does not change the solutions to the system of equations 𝐴π‘₯ = 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 + π‘₯ where π‘₯ ∈ 𝑁(𝐴).
33
System of linear equations is 'consistent'?
either one solution or infinitely many solutions.
34
System of linear equations is 'inconsistent'?
A system of linear equations is said to be inconsistent if it has no solution
35
Solving homogeneous linear system what are free variables?
The non-pivot, non-zero variables in a matrix that is in reduced echelon form
36
Formula for number of free variables?
num. free var = num. var - num. pivots = num. var - num. constraints
37
Dot product a β‹… b = b β‹… a = a scalar = magnitude of a * magnitude of b * cosine of angle between them
38
Cross product a x b /= b x a = magnitude of a * magnitude of b * sine of angle between them unit vector, perpendicular to both a and b (use right hand rule to know which way)