Test 1 Review PDF

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Test 1 Review, Teacher: Akram Aldroubi...

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Math 2500 Review Test 1 1

Math 2500: Review for Test One

Our first test will be Thursday, 19 September 2019 (during the TA session and in the regular Thursday classroom). It covers the course notes on Brightspace, Chapter 1, Sections 1 through 4 (pages 1—43) of Shifrin, Collected Homework 1, 2, 3, 4, lectures, and discussions. The test will be 50 minutes. Calculators (or other electronic equipment) are not allowed. Definitions, formulas, etc. to know: 1. mathematical induction 2. algebra of vectors (scalar multiplication and vector addition) 3. Parallelogram law of vector addition xi

4. length of vector

∑¿ ¿ ¿

¿ √¿ 5. unit vector 6. dot product: x ∙ y=x 1 y 1 +…+ xk y k (the sum of all the components) 7. Cauchy-Schwarz inequality: |x ∙ y|≤||x ||∗¿|  y |∨¿ and if |x ∙y |=| x |∗¿| y|∨¿ , then x and y are parallel. ¿ |y|∨¿ Proof:

y ¿|x |∨¿− ¿

 z 2> 0

Assume a function z=

do the algebra to just go through.

x ¿

8. the angle between two vectors:

||x||∗¿| y|∨¿ cos ( θ )=

x ∙  y ¿

9. orthogonality of vectors 10. Triangle Inequality: 11. projection of x

||x +y||≤||x|+¿ |y |∨¿

onto y :

pro jy ( x ) =c y =( ||x||∗cos ( θ ) )∗y=

x ∙ y

||y||2

∗y

12. standard basis vectors 13. linear subspaces: contain 0 and closed under addition and scaler multiplication 14. affine subspaces: a linear subspace that has another vector added to it. All subspaces are affine but not every affine is a subspace. 15. abstract vector spaces

2

16. linear combinations: any combination of vectors with a scaler multiple 17. span of vectors: span (x 1 … x k )=c 1 x 1 +…+c k v k

for all c ∈ R

18. orthogonal subspaces 19. orthogonal complement 20. linear transformation: Fulfills the 2 linear transformation axioms: closed under addition and scaler multiplication 21. Rotations in R 2: Rθ is a linear transformation (addition and scaler multiplication). Rθ=

[

cos(θ) −sin (θ) sin (θ) cos(θ)

[ [ [

]

1 0 0 R rotation about x−axis 0 cos(θ) −sin (θ) 0 sin (θ) cos(θ) 3

R3 rotation about y−axis

. Rotates a vector by an angle of θ

]

] ]

cos ( θ ) 0 sin ( θ) 0 1 0 −sin ( θ ) 0 cos( θ )

cos ( θ ) −sin ( θ ) 0 R3 rotation about z−axis sin ( θ ) cos ( θ ) 0 0 0 1

22. Matrices: algebraic operations, especially multiplication:

][

][

u v A∗B= a b c ∗ w x = row 1 of A doted into column1 of B row 1 of A dottted into column 2 of B d e f row 2 of A doted into column1 of B row 2 of A doted into column 2 of B y z

[

Also… S ∘T =[ S] [T ] 23. Mm,n is an abstract vector space 24. Standard matrix [T] of a linear transformation T 25. Invertible matrices and inverses: Matrix are invertible if determinant ≠ 0 26. Explicit formula for the inverse of an invertible 2 x 2 matrix: 1 ∗ d −b det(A ) −c a

[

]

−1 −1 −1 27. The shoe-sock theorem: ( AB ) =B A

28. Transposes and their properties: The transpose, B, of a matrix A is where [bij ]=[ a ji ] 29. Ax · y = x · ATy 30. 2 x 2 determinant

]

3

31. symmetric and skew-symmetric matrices 32. orthogonal matrices and transformations:...