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Unformatted text preview: Vector space and linear combination
2-D example to illustrate the reach of a linear combination of 2 vectors
• Span of the 2 vectors (different cases) • Subspace (a line or a plane?) 3-D example
• The span of 2 vectors • Subspace (a line or a plane?) 13 13 Linear dependence (LD) and linear independence (LI)
LD and LI are related with respect to a LD given set of vectors LD means any vector of this set is related LD to the other vectors through a linear combination relationship; otherwise LI A conceptual link to relate vectors with conceptual linear equations 14 LI, LD, and homogeneous linear equations
LI implies a unique (and trivial) solution LI to a homogeneous linear equation set (Ax=0) LD implies an infinity of solutions LD (including the trivial solution x=0) to Ax=0 15 15 Solutions to Ax=y and Ax=0
Use 2 equations set to explain the Use situations for 2 lines Three solution cases for Ax=y Three (inconsistent, unique, and infinity) Two solution cases for Ax=0 (unique Two and trivial, infinity) 16 Linear independence and homogenous equation (LI and Ax=0)
The linear combination equation can be The interpreted as Ax=0 To prove LI, we need to show the To uniqueness of the trivial solution (x=0) Gauss elimination is useful for proving Gauss uniqueness
• 2-D and 3-D examples 17 17 Gauss elimination for Ax=0
Explore the structure of [Ux |0] after Explore forward substitution Starting from the last row and perform Starting interpretation row by row
• What would you conclude on alpha if alpha*0 = 0? • What would you conclude on alpha if alpha*k =0, and k is non-zero?
18 3-D example of 2 vectors
Given 2 vectors (1,2,2) and (-1,0,2), find Given (a) the span and (b) test whether they are LI Draw perspective view and interpret Draw without using numbers
• Span should be a plane (why?) • The two vectors are LI. It is simply not possible to obtain one vector from the other using linear combination
19 19 Using Gauss elimination for previous example
(a) Finding span (a)
• Define a point within the span as (x,y,z) and construct linear equation set • Apply Gauss elimination and use consistency for the last row (b) Testing for LI (b)
• Construct homogeneous equation set • Apply Gauss elimination to verify solution uniqueness
20 Generalization of LI and LD concepts
A set of n-dimensional vectors set The same concepts on linear The combination, LI, and LD still apply The Gauss elimination procedure The becomes more complicated because of the size of the A matrix in considering Ax=y and Ax=0
• Try to use row interchange to get more zeros in the lower rows
21 21 Basis of a vector space
A set of LI vectors which span the set vector space Any vector in the vector space can be Any expanded uniquely by a linear combination Previous 3-D example: The two vectors Previous form a basis for the vector space (plane)
22 Standard bases for R2 and R3
2-D example is so familiar (i and j) Orthogonal and orthonormal basis Orthogonal
• Perpendicular vectors • Normalized vectors 3-D example (i,j,k) and n-dimension For any vector, finding the linear For combination is simple on these bases – coordinates?
23 23 Linear combination using orthogonality and ortho-normality
Projection: How to find the components Projection: based on orthogonality and orthonormality (ON)? Finding the vector components can be Finding related to our interpretation of physical phenomenon and even behaviors
• Satellites operation or just running or cycling in circles
24 Gram-Schmidt process
Obtaining k orthonormal (ON) vectors Obtaining from k linearly independent (LI) vectors Verifying the correctness of the process Verifying
• Each of the k ON vectors can be obtained as a linear combination of the k LI vectors • Ortho-normality holds for the k ON vectors 25 25 Dimension
How should we interpret this word? How There are many dimensions to this problem … There It is a 3-dimensional space … Let’s consider space-time as 4 dimensions … How to represent the information of this economic problem as an n-dimensional vector k-dimensional vector space is a vector space in which the greatest number of LI vectors is k 26 ...
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This note was uploaded on 11/13/2010 for the course SEEM SEEM2018 taught by Professor Chan during the Spring '10 term at CUHK.
- Spring '10