Frames Vectors and Vector Spaces Orthonormal Loosen Up! An introduction to frames. Bases Frames 4 (Interrelated) Research Areas Keri A. Kornelson Applied Math Linear Algebra Geometry University of Oklahoma - Norman Operator Theory [email protected] Frames for Undergraduates Joint AMS/MAA Meetings Panel: This could be YOUR graduate research! New Orleans, LA January 7, 2011 Quick review of vector spaces, Rn: Frames Vectors and Vector Spaces Orthonormal Vectors in Rn are sometimes represented as columns: Bases Frames x(1) 4 (Interrelated) x(2) Research Areas x Applied Math = . Linear Algebra . Geometry Operator Theory x(n) Frames for Undergraduates Quick review of vector spaces, Rn: Frames n Vectors and Vector Vectors in R are sometimes represented as columns: Spaces Orthonormal x(1) Bases x(2) Frames x = . 4 (Interrelated) . Research Areas Applied Math x(n) Linear Algebra Geometry Operator Theory Frames for and sometimes as arrows: Undergraduates Norms and Dot Products Frames 2 vectors x(1) y(1) Vectors and Vector x(2) y(2) Spaces x y = . = . Orthonormal . Bases x(n) y(n) Frames 4 (Interrelated) Definition Research Areas Applied Math The norm or length of x is Linear Algebra Geometry 1 Operator Theory n 2 Frames for 2 kxk = x(i) . Undergraduates = ! Xi 1 A vector with norm 1 is called a unit vector . Norms and Dot Products Frames 2 vectors x(1) y(1) Vectors and Vector x(2) y(2) Spaces x y = . = . Orthonormal . Bases x(n) y(n) Frames 4 (Interrelated) Definition Research Areas Applied Math The norm or length of x is Linear Algebra Geometry 1 Operator Theory n 2 Frames for 2 kxk = x(i) . Undergraduates = ! Xi 1 A vector with norm 1 is called a unit vector . Definition The dot product or inner product of x and y is n hx, yi = x(i)y(i). = Xi 1 Orthogonal Vectors Frames Vectors and Vector Definition Spaces Orthonormal Two vectors are orthogonal if their inner (dot) product is zero, Bases i.e. if their “arrows” are perpendicular. Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthogonal Vectors Frames Definition Vectors and Vector Spaces Two vectors are orthogonal if their inner (dot) product is zero, Orthonormal i.e. if their “arrows” are perpendicular. Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Definition Two vectors are orthonormal if they are orthogonal and are both unit vectors. Orthonormal Bases Frames Vectors and Vector Definition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames Some handy facts about ONBs: 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases Frames Vectors and Vector Definition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames Some handy facts about ONBs: 4 (Interrelated) Research Areas I Applied Math The unique expansion coefficients are found by the dot Linear Algebra Geometry product. Operator Theory n n Frames for x = ci bi = hx, bi ibi Undergraduates = = Xi 1 Xi 1 Orthonormal Bases Frames Vectors and Vector Definition Spaces n A collection of vectors {bi }i=1 is an orthonormal basis (ONB) Orthonormal for Rn if the vectors are pairwise orthonormal and form a basis. Bases Frames Some handy facts about ONBs: 4 (Interrelated) Research Areas I Applied Math The unique expansion coefficients are found by the dot Linear Algebra Geometry product. Operator Theory n n Frames for x = ci bi = hx, bi ibi Undergraduates = = Xi 1 Xi 1 I Parseval’s Identity: n 2 2 kxk = |hx, bi i| = Xi 1 A Silly Example Frames Vectors and Vector Spaces Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0 4 (Interrelated) 0 0 1 Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates A Silly Example Frames Vectors and Vector Spaces Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0 4 (Interrelated) 0 0 1 Research Areas Applied Math Linear Algebra Geometry 4 Operator Theory x = 5 = 4b1 + 5b2 + 6b3 Frames for 6 Undergraduates A Silly Example Frames Vectors and Vector Spaces Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0 4 (Interrelated) 0 0 1 Research Areas Applied Math Linear Algebra Geometry 4 Operator Theory x = 5 = 4b1 + 5b2 + 6b3 Frames for 6 Undergraduates hx, b1i = 4, hx, b2i = 5, hx, b3i = 6. A Silly Example Frames Vectors and Vector Spaces Orthonormal 1 0 0 Bases Frames b1 = 0 b2 = 1 b3 = 0 4 (Interrelated) 0 0 1 Research Areas Applied Math Linear Algebra Geometry 4 Operator Theory x = 5 = 4b1 + 5b2 + 6b3 Frames for 6 Undergraduates hx, b1i = 4, hx, b2i = 5, hx, b3i = 6. kxk2 = 42 + 52 + 62 p Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal I All vectors have norm 1. Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math I All the vectors are pairwise orthogonal. Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math I All the vectors are pairwise orthogonal. Linear Algebra Geometry Operator Theory Frames for Undergraduates Orthonormal Bases, cont. Frames Vectors and Vector ONB’s are pretty restrictive...they all look alike somehow. Spaces Orthonormal I All vectors have norm 1. Bases I The number of vectors n equals the dimension of the Frames Rn 4 (Interrelated) space: . Research Areas Applied Math I All the vectors are pairwise orthogonal. Linear Algebra Geometry Operator Theory Frames for Undergraduates I There’s not much flexibility to tailor an ONB to a particular application, and there is no resilience to losses or errors in data reconstruction. Signal transmission Frames n If we agree on an ONB {bi } = , then I can just send you the Vectors and Vector i 1 Spaces coefficients of x and you can find x. Orthonormal In our silly example, I send you 4, 5, 6 and you can compute Bases Frames 4 4 (Interrelated) Research Areas x = 4b1 + 5b2 + 6b3 = 5 . Applied Math Linear Algebra 6 Geometry Operator Theory Frames for Same idea works for voice on a cell phone or pictures sent Undergraduates over the internet. Signal transmission Frames n If we agree on an ONB {bi } = , then I can just send you the Vectors and Vector i 1 Spaces coefficients of x and you can find x. Orthonormal In our silly example, I send you 4, 5, 6 and you can compute Bases Frames 4 4 (Interrelated) Research Areas x = 4b1 + 5b2 + 6b3 = 5 . Applied Math Linear Algebra 6 Geometry Operator Theory Frames for Same idea works for voice on a cell phone or pictures sent Undergraduates over the internet. If one data point gets lost using an ONB, there is no information about what it was. 4 4, ?, 6 −→ 4b1 + ?b2 + 6b3 = ? 6 Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces Orthonormal Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces Orthonormal Stretch out Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces Orthonormal Stretch out Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Move around a bit Operator Theory Frames for Undergraduates Loosen up! Frames The concept of a frame for a vector space allows for more Vectors and Vector wiggle room than ONBs. The vectors are allowed to Spaces Orthonormal Stretch out Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Move around a bit Operator Theory Frames for Undergraduates Even invite a few friends over! Frames Frames n Vectors and Vector 2 2 Recall Parseval’s Identity for ONB {bi }: kxk = |hx, bi i| Spaces i=1 Orthonormal X Bases Frames 4 (Interrelated) Research Areas Applied Math Linear Algebra Geometry Operator Theory Frames for Undergraduates Frames Frames n Vectors and Vector 2 2 Recall Parseval’s Identity for ONB {bi }: kxk = |hx, bi i| Spaces i=1 Orthonormal X Bases Definition Frames Rn k A frame for is a collection of vectors {fi }i=1 that satisfy a 4 (Interrelated) looser condition than Parseval’s identity.