Your SlideShare is downloading. ×
0
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Linear Algebra
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Linear Algebra

1,500

Published on

Linear Algebra

Linear Algebra

Published in: Education, Technology
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,500
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
70
Comments
0
Likes
2
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. 6.837 Linear Algebra Review Patrick Nichols Thursday, September 18, 2003 6.837 Linear Algebra Review
  • 2. Overview <ul><li>Basic matrix operations (+, -, *) </li></ul><ul><li>Cross and dot products </li></ul><ul><li>Determinants and inverses </li></ul><ul><li>Homogeneous coordinates </li></ul><ul><li>Orthonormal basis </li></ul>6.837 Linear Algebra Review
  • 3. Additional Resources <ul><li>18.06 Text Book </li></ul><ul><li>6.837 Text Book </li></ul><ul><li>[email_address] </li></ul><ul><li>Check the course website for a copy of these notes </li></ul>6.837 Linear Algebra Review
  • 4. What is a Matrix? <ul><li>A matrix is a set of elements, organized into rows and columns </li></ul>6.837 Linear Algebra Review rows columns
  • 5. Basic Operations <ul><li>Addition, Subtraction, Multiplication </li></ul>6.837 Linear Algebra Review Just add elements Just subtract elements Multiply each row by each column
  • 6. Multiplication <ul><li>Is AB = BA? Maybe, but maybe not! </li></ul><ul><li>Heads up: multiplication is NOT commutative! </li></ul>6.837 Linear Algebra Review
  • 7. Vector Operations <ul><li>Vector: 1 x N matrix </li></ul><ul><li>Interpretation: a line in N dimensional space </li></ul><ul><li>Dot Product, Cross Product, and Magnitude defined on vectors only </li></ul>6.837 Linear Algebra Review x y v
  • 8. Vector Interpretation <ul><li>Think of a vector as a line in 2D or 3D </li></ul><ul><li>Think of a matrix as a transformation on a line or set of lines </li></ul>6.837 Linear Algebra Review V V’
  • 9. Vectors: Dot Product <ul><li>Interpretation: the dot product measures to what degree two vectors are aligned </li></ul>6.837 Linear Algebra Review A B A B C A+B = C (use the head-to-tail method to combine vectors)
  • 10. Vectors: Dot Product 6.837 Linear Algebra Review Think of the dot product as a matrix multiplication The magnitude is the dot product of a vector with itself The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle
  • 11. Vectors: Cross Product <ul><li>The cross product of vectors A and B is a vector C which is perpendicular to A and B </li></ul><ul><li>The magnitude of C is proportional to the cosine of the angle between A and B </li></ul><ul><li>The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system” </li></ul>6.837 Linear Algebra Review
  • 12. Inverse of a Matrix <ul><li>Identity matrix: AI = A </li></ul><ul><li>Some matrices have an inverse, such that: AA -1 = I </li></ul><ul><li>Inversion is tricky: (ABC) -1 = C -1 B -1 A -1 </li></ul><ul><li>Derived from non-commutativity property </li></ul>6.837 Linear Algebra Review
  • 13. Determinant of a Matrix <ul><li>Used for inversion </li></ul><ul><li>If det(A) = 0, then A has no inverse </li></ul><ul><li>Can be found using factorials, pivots, and cofactors! </li></ul><ul><li>Lots of interpretations – for more info, take 18.06 </li></ul>6.837 Linear Algebra Review
  • 14. Determinant of a Matrix 6.837 Linear Algebra Review Sum from left to right Subtract from right to left Note: N! terms
  • 15. Inverse of a Matrix 6.837 Linear Algebra Review <ul><li>Append the identity matrix to A </li></ul><ul><li>Subtract multiples of the other rows from the first row to reduce the diagonal element to 1 </li></ul><ul><li>Transform the identity matrix as you go </li></ul><ul><li>When the original matrix is the identity, the identity has become the inverse! </li></ul>
  • 16. Homogeneous Matrices <ul><li>Problem: how to include translations in transformations (and do perspective transforms) </li></ul><ul><li>Solution: add an extra dimension </li></ul>6.837 Linear Algebra Review
  • 17. Orthonormal Basis <ul><li>Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis </li></ul><ul><li>Ortho-Normal: orthogonal + normal </li></ul><ul><li>Orthogonal: dot product is zero </li></ul><ul><li>Normal: magnitude is one </li></ul><ul><li>Example: X, Y, Z (but don’t have to be!) </li></ul>6.837 Linear Algebra Review
  • 18. Orthonormal Basis 6.837 Linear Algebra Review X, Y, Z is an orthonormal basis. We can describe any 3D point as a linear combination of these vectors. How do we express any point as a combination of a new basis U, V, N , given X, Y, Z ?
  • 19. Orthonormal Basis 6.837 Linear Algebra Review (not an actual formula – just a way of thinking about it) To change a point from one coordinate system to another, compute the dot product of each coordinate row with each of the basis vectors.
  • 20. Questions? <ul><li>? </li></ul>6.837 Linear Algebra Review

×