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- 1. 1 M01 L01: Pre-requisites 2. Mathematica 1. GeoGebra 3. MATLAB GeoGebra is an interactive geometry, algebra, statistics and calculus application and available on multiple platforms, with apps for desktops, tablets and web. It’s a freeware and many of the assignments will be in GeoGebra. Mathematica and MATLAB will be useful for understanding the Linear Algebra and Calculus in more interactive way. IIT Guwahati provide licenses of these software and Many of the assignments will be in MATL5AB and Mathematica. YOU HAVE TO USE THESE SOFTWARE WITHOUT ANY EXCUSE
- 2. M01: Linear Algebra, L01: Fundamentals of Vectors Course Instructor Dr. Sajan Kapil Department of Mechanical Engineering Indian Institute of Technology, Guwahati Guwahati, Assam ME 501 IIT Guwahati Advanced Engineering Mathematics (3−0−0−6)
- 3. 3 M01 L01: Contents Perspective of Vectors Geometry of Vectors Vector as Arrow Vectors as Coordinates Vectors in 𝑹𝒏 Fundamental Vector Operations Vector Addition Scalar Multiplication Basis Vectors Linear Combination of Vectors Linear Span of 2D Vectors Linear Span of 3D Vectors
- 4. 4
- 5. 5 M01 L01: Geometry of Vector A vector is a term that refers colloquially to some quantities that cannot be expressed by a single number, OR to elements of some vector spaces. Displacement, Velocity, Acceleration, Force, Momentum A vector quantity is defined as the physical quantity that has both directions as well as magnitude. Vector as Arrow A B X Y O Tail Head A B X Y O Z 𝐴𝐵 vector in 2D Coordinate system Or XY Cartesian plane 𝐴𝐵 vector in 3D Coordinate system
- 6. 6 M01 L01: Perspective of Vector 𝐕 𝑽𝒙 𝑽𝒚 𝑽𝒛 Physics Student Math's Student Computer Science Student The mathematics perspective is more ABSTRACT. A vector space over a field F is a set V together with two binary operations that satisfy the eight axioms. The elements of V are commonly called vectors, and the elements of F are called scalars. This perspective will be considered after a few lectures. 2D Arrow or 3D Arrows Ordered list of Numbers
- 7. 7 M01 L01: Vector as Coordinates It is also obvious that the 3D vectors can be also be represented in the 3D coordinate system as: 𝐚 = 𝑶𝑨 = 𝒂𝒙 𝒂𝒚 𝒂𝒛 𝟓 𝟓 𝟏𝟎
- 8. 8 M01 L01: Vector as Arrow A B X Y O X Y O A 𝐚 𝑂𝐵 𝑂𝐹 𝑂𝐷 𝑂𝐻 𝑂𝐸 𝑂𝐼 𝑂𝐺 𝑂𝐶 Origin of the 2D Coordinate system Standard Vector: with tail sitting on the origin
- 9. 9 M01 L01: Vector as Coordinates X Y ax ay O A 𝐚 𝑂𝐵 𝑂𝐹 𝑂𝐷 𝑂𝐻 𝑂𝐸 𝑂𝐼 𝑂𝐺 𝑂𝐶 𝐚 = 𝑶𝑨 = [𝐚𝐱, 𝐚𝐲] 𝐚𝐱, 𝐚𝐲 ≠ 𝒂𝒚, 𝒂𝒙 The individual coordinate of the vector is called as component. Vectors may also be called as ordered list of numbers. It is also more convenient to write the components in column instead of row. This helps in further computation of the vectors. 𝐚 = 𝑶𝑨 = 𝒂𝒙 𝒂𝒚 and obviously: 𝒂𝒙 𝒂𝒚 ≠ 𝒂𝒚 𝒂𝒙 𝐛 = 𝟓 𝟒 𝐟 = 𝟑 𝟕 𝐝 = −𝟒 𝟓 𝐡 = −𝟓 𝟐 𝐄 = −𝟏 −𝟏 𝐈 = −𝟏 −𝟒 𝐠 = 𝟑 −𝟒 𝐠 = 𝟔 −𝟑
- 10. 10 M01 L01: Vectors in ℝ𝑛 In order to first understand the illustration of Linear Algebra, we shall keep the current discussion limited to ℝ𝟐 and ℝ𝟑 and extend the same analogy after a few lectures.
- 11. 11 M01 L01: Vector (Operations) Linear algebra revolve around two fundamental operations: 1. Vector Addition 2. Scalar multiplication This is an abstract perspective of mathematicians and hence will be explained after a few lectures. Why only these two operations are chosen by mathematicians ?
- 12. 12 M01 L01: Vector Addition y x 𝑨 + 𝑩 𝑨 𝑩
- 13. 13 M01 L01: Vector Addition y x 𝑨 + 𝑩 𝑨 𝑩 𝑪 Why moving a vector in the space like this is correct ?
- 14. 14 M01 L01: Vector Addition y x 𝑨 + 𝑩 𝑨 Why this method of vector addition is wrong ? Wrong 𝑩 𝑪
- 15. 15 M01 L01: Vector Addition y x 3 3 + 6 −4 9 −1 3 3 -1+4=3 1 4 -1+4=? 1 4 1+4=? 1+4=5 Do you have answers to the last two question ? (3 − 4) (3 + 6) 𝑨 𝑩 𝑪 𝑨 + 𝑩 3 3 + 6 −4 = 3 + 6 3 − 4 = 9 −1 𝑨 + 𝑩 = 𝑪 𝒙𝟏 𝒚𝟏 + 𝒙𝟐 𝒚𝟐 = 𝒙𝟏 + 𝒙𝟐 𝒚𝟏 + 𝒚𝟐 -4 6
- 16. 16 Vector Scalar Multiplication y x 𝑨 −𝟐𝑨 Scaling Scalars 𝟐𝑨 𝒂𝒙 𝒂𝒚 𝟐𝒂𝒙 𝟐𝒂𝒚 −𝟐𝒂𝒚 −𝟐𝒂𝒙 Multiplying with numbers is basically flipping and scaling a vector. Hence this number is called as scalar. Numerically: 𝑐𝐴 = c 𝑎𝑥 𝑎𝑦 = 𝑐 × 𝑎𝑥 𝑐 × 𝑎𝑦 So we have understood the two fundamental operations in vectors i.e. scalar multiplication and vector addition.
- 17. 17 Vector: Basis Vectors y x (𝟓) 𝑨 (−𝟖) 5 −8 𝑖 𝑗 5𝑖 −8𝑗 𝑨 = (𝟓) × 𝒊 + −𝟖 × 𝒋 Scalar multiplication Vector Addition If there are two vectors 𝑖 and 𝑗 with a unit length. Vector 𝑖 is along the 𝑥-axis and vector 𝑗 is along the 𝑦-axis. Then any vector in the 𝑥𝑦 plane can be represented by (a linear combination of) using 𝑖 and 𝑗 vector with appropriate scalar multiplication and vector addition. 𝑖 and 𝑗 are called as basis vectors of the 𝒙𝒚 coordinate system. Can we represent any vector in the 𝑥𝑦 plane using the aforesaid two operations and two fundamental vectors ?
- 18. 18 Vector: Basis Vectors 𝑣 𝑤 The answer is yes but its not a standard way of doing it as one may not choose same basis vectors and hence we can not be on the same page. Can we choose some different Basis vectors and still able to define each vector in the plane by (a linear combination of) using those different basis vectors (𝒗 and 𝒘) with two operations viz., scalar multiplication and vector addition. If so, it will be a new coordinate system. 𝑣 𝑢 P 2𝑣 1.5𝑤 𝑣 + 𝑤 2𝑣 + 1.5𝑤 1 1 2 1.5
- 19. 19 Vector: Basis Vectors 𝑣 𝑤 The answer is yes but its not a standard way of doing it as one may not choose same basis vectors and hence we can not be on the same page. Can we choose some different Basis vectors and still able to define each vector in the plane by (a linear combination of) using those different basis vectors (𝒗 and 𝒘) with two operations viz., scalar multiplication and vector addition. If so, it will be a new coordinate system. 2𝑣 1.5𝑤 𝑣 + 𝑤 2𝑣 + 1.5𝑤 1 1 2 1.5 𝑖 𝑗 10𝑖 + 4.5𝑗 10 4.5 ≠ Just like: 𝟏𝟎𝟏𝟐 ≠ 𝟏𝟎𝟏𝟏𝟎 So, in this course, if you see a vector 𝑎𝑥 𝑎𝑦 and no specific base of the vector is mentioned then you will consider 𝑥 and 𝑦 axes as base vector 𝑣 𝑢 P
- 20. 20 Paradox: Statement: ‘101’ is a number in base ‘10’ Is the base of this number ‘10’ Is the base of this number ‘10’ Is the base of this number is ‘10’ Importance of Basis Vectors Vector: Basis Vectors
- 21. 21 Vector: Linear combination In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by an scalar and adding the results (e.g. a linear combination of x and y would be an expression of the form of 𝒂𝒙 + 𝒃𝒚, where a and b are scalar). Term Vector 𝒂 𝒗 + 𝒃𝒖 This is a vector which is a linear combination of vectors 𝑣 & 𝑢 Scalars
- 22. 22 Vector: Linear combination Determine whether 4 −1 is a linear combination of 2 3 and 3 1 4 −1 = x1 2 3 + x2 3 1 If 4 −1 is a linear combination of 2 3 and 3 1 then: 4 −1 = 2x1 + 3x2 3𝑥1 + 𝑥2 2x1 + 3x2 = 4 3𝑥1 + 𝑥2 = −1 2x1 + 3x2 = 4 −9𝑥1 − 3𝑥2 = 3 −7x1 = 7 𝒙𝟏 = −𝟏 𝒙𝟐 = 𝟐 4 −1 = −1 × 2 3 + 2 × 3 1 It’s a unique solution hence 4 −1 is a linear combination of 2 3 and 3 1
- 24. 24 Vector: Linear combination Determine whether −4 −2 is a linear combination of 6 3 and 2 1 −4 −2 = x1 6 3 + x2 2 1 If −4 −2 is a linear combination of 6 3 and 2 1 then: −4 −2 = 6𝑥1 + 2𝑥2 3𝑥1 + 𝑥2 6𝑥1 + 2𝑥2 = −4 3𝑥1 + 𝑥2 = −2 3x1 + x2 = −2 Any combination of 𝑥1and 𝑥2which satisfy the 3x1 + x2 = −2 is a solution, hence there are infinite solutions. One such solution can be: 𝑥1 = 2 & 𝑥2 = 4, so −4 −2 = −2 × 6 3 + 4 × 2 1 Therefore: −4 −2 is a linear combination of 6 3 and 2 1
- 26. 26 Vector: Linear combination Determine whether 1 2 is a linear combination of 2 3 and 6 9 1 2 = x1 2 3 + x2 6 9 If 1 2 is a linear combination of 2 3 and 6 9 then: 1 2 = 2𝑥1 + 6𝑥2 3𝑥1 + 9𝑥2 2𝑥1 + 6𝑥2 = 1 3𝑥1 + 9𝑥2 = 2 Here, 1 3 = 0, hence there are no solutions. Therefore: 1 2 is NOT a linear combination of 2 3 and 6 9 𝑥1 + 𝑥2 = 1/3 𝑥1 + 𝑥2 = 2/3
- 28. 28 Vector: Linear combination Write the vector 1 2 3 as a linear combination of the vectors: 1 0 1 , 1 1 0 & 0 1 1
- 29. 29 Vector: Linear Span 𝒂 𝒗 + 𝒃𝒖 The span of vector 𝒗 & 𝒖 is the set of all the linear combinations of vectors 𝒗 & 𝒖 𝑣 𝑢 P So, basically if 𝑣 and 𝑢 are linearly independent (𝑣 ≠ 𝑐𝑢) then the span of 𝑣 and 𝑢 will be the entire 2D space. However, if 𝑣 and 𝑢 are linearly dependent (𝑣 = 𝑐𝑢) then the span of 𝑣 and 𝑢 will be the entire a line. 𝑣 𝑢 = 𝑐𝑣 𝑖 𝑗
- 30. 30 Vector: Linear Span 𝒂 𝒗 + 𝒃𝒖 So, basically if 𝑣 and 𝑢 are linearly independent (𝑣 ≠ 𝑐𝑢) then the span of 𝑣 and 𝑢 will be the entire 2D space.
- 31. 31 if 𝑣 and 𝑢 are linearly dependent (𝑣 = 𝑐𝑢) then the span of 𝑣 and 𝑢 will be the entire a line. Vector: Linear Span 𝒂 𝒗 + 𝒃𝒖
- 32. 32 Vector: Linear Span It a bad idea to show the span of two vectors (which is a set all the 2D vectors for linearly independent vectors) to show by ARROWS. Hence it is the time to represent them by points.
- 36. 36 Vector: Linear Span 𝒂𝒗 + 𝒃𝒖 + 𝒄𝒘 Case 1: 𝒗, 𝒖 and 𝒘 are linearly independent Demo in Solidworks & GeoGebra
- 38. 38 Vector: Linear Span 𝒂𝒗 + 𝒃𝒖 + 𝒄𝒘 Case 2: 𝒗 is independent to 𝑢, but 𝑢 are 𝑤 linearly dependant Demo in Solidworks & GeoGebra
- 39. 39 Vector: Linear Span 𝒂𝒗 + 𝒃𝒖 + 𝒄𝒘 Case 3: 𝑣, 𝑢, and 𝑤 are linearly dependent Demo in Solidworks & GeoGebra
- 40. 40 Vector: Basis Vectors The basis of vector space is a set of linearly independent vectors that span the full space.
- 41. Thank You Course Instructor Dr. Sajan Kapil Department of Mechanical Engineering Indian Institute of Technology, Guwahati Guwahati, Assam ME 501 IIT Guwahati Advanced Engineering Mathematics (3−0−0−6)