Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
M. van der Weiden
Week 1
melanie.vanderweiden@Inholland.nl
7 september 2022
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Linear Algebra
Learning material
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Linear Algebra
Subjects of this course
Introduction to matrices
Matrix operations
Vector multiplication
Gaussian elimination
Determinants
Inverse matrix and its properties
Singular and non-singular matrices
Solving system of simultaneous equations
Rank
Eigenvalues and eigenvectors
Linear transformations:
Reflection matrix
Rotation matrix
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Linear Algebra
Why linear algebra?
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Linear Algebra
Linear algebra
note: 1 means leaves the node, -1 means enters the node, 0 no connection
between nodes
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Linear Algebra
Introduction to matrices
A system of linear equations:
x1 − 2x2 = −1
−x1 + 3x2 = 3
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Linear Algebra
Introduction to matrices
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Linear Algebra
Dimension of a matrix
a11 . . . a1m
A = a21
..
. a2m
an1 . . . anm
The size of a matrix is the number of rows times the number of columns
Dim(A) = n × m. A square matrix has dimension n × n.
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Linear Algebra
Special matrices
Identity matrix
1 0
I=
0 1
This is also a diagonal matrix.
Inverse of the matrix A is denoted A−1 .
The transpose AT , rows become columns.
Upper triangular matrix / echelon form
a11 a12 a13
A = 0 a22 a23
0 0 a33
1 8
3 2
A=
4
3
2 4
then
T 1 3 4 2
A =
8 2 3 4
Dim(A) = 4 × 2 and Dim(AT ) = 2 × 4
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Linear Algebra
Matrix operations
Addition and subtraction: Dimensions need to be the same
Example addition: add a11 + b11 and do that will all the coefficients.
Add
1 8
3 2
A= 4 3
2 4
and
1 −5
−2 6
B=
3 −7
4 8
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Linear Algebra
Matrix operations
Example addition:
1 8 1 −5 2 3
+ −2 6 = 1 8
3 2
4 3 3 −7 7 −4
2 4 4 8 6 12
Now do the subtraction.
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Linear Algebra
Matrix operations
2 4 4 8
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Linear Algebra
Matrix operations
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Linear Algebra
Matrix operations
and
3 −1
B = 4 2
2 0
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Linear Algebra
Matrix operations
Multiply matrix A with matrix B example.
1 2 −3
A = 5 0 2
1 −1 −1
and
3 −1
B = 4 2
2 0
Yields:
5 3
A · B = 19 −5
−3 −3
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Linear Algebra
Properties of matrix multiplication
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Linear Algebra
Exercise 1
and
1 1 0
D=
2 0 1
And what is DC?
Is CD = DC?
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Linear Algebra
Exercise 2
Don’t forget that the dimensions are very important! Row × column.
What is the result of the multiplication of AB:
2 −3 8 4
A= ,B =
−4 6 5 5
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Linear Algebra
Vectors
a
b
Column vector v =
. . .
n
Row vector a = a1 a2 . . . an
Notation of a vector: ~a, a or a
The row vector is the transpose of the column vector vT = a
Unit vector has a length of 1, notation v̂
Scalar product or dot product (inproduct)
Vector product or cross product (uitproduct)
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Linear Algebra
Scalar product
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Linear Algebra
Cross product
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Linear Algebra
Cross product
This is used when you want to find a Moment.
M=r×F
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Linear Algebra
Cross product
There are different ways to calculate the vector product:
Method 1:
i j k
a2 a3
a1 a3
a1 a2
a2 b3 − b2 a3
b2 b3 i − b1 b3 j + b1 b2 k = b1 a3 − a1 b3
a1 a2 a3 =
b1 b2 b3 a1 b2 − b1 a2
Method 2: starting point now make 3 crosses.
a1 b1
a2 b2
a3 b3
a1 b1
a2 b2
a3 b3
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Linear Algebra
Cross product exercise
7 1
a = 1 and b = 3 calculate the crossproduct.
2 −2
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Linear Algebra
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Linear Algebra
Course schedule week 1
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Linear Algebra
Answer subtraction
0 13
5 −4
A−B =
1 10
−2 −4
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Linear Algebra
Answer exercise 1
1 −2
1 1 0 1 · 1 + 1 · −1 + 0 · −2 1 · −2 + 1 · 2 + 0 · 4
DC = −1 2 =
2 0 1 1 · 2 + 0 · −1 + 1 · −2 2 · −2 + 0 · 2 + 1 · 4
−2 4
0 0
=
0 0
−3 1 2
CD = 3 −1 2
6 −2 4
DC 6= CD
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Linear Algebra
Answer exercise 2
2 −3 8 4 8 · 2 + −3 · 5 2 · 4 + −3 · 5 1 −7
AB = = =
−4 6 5 5 8 · −4 + 6 · 5 4 · −4 + 6 · 5 −2 14
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Linear Algebra
Answer exercise 3
−8
a × b = 16
20
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