102 - unique linear combination What is the condition in R...

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1 Example: Given the coefficients ({ 3,4,1}), it is easy to compute the combination ([2 8] T ), but the inverse problem is harder. Example: To determine x 1 and x 2 , must solve a system of linear equations, which has a unique solution [ x 1 x 2 ] T = [ 1 2] T in this case. Geometrical view point: manage to form a parallelogram Example: to determine if [ 4 2] T is a linear combination of [6 3] T and [2 1] T , must solve which has infinitely many solutions , as the geometry suggests.
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2 Example: to determine if [3 4] T is a linear combination of [3 2] T and [6 4] T , must solve which has no solutions , as the geometry suggests. Proposition: If w 1 , w 2 , , w p are all linear combinations of the vectors u 1 , u 2 , , u l , then any linear combinations of w 1 , w 2 , , w p is still a linear combinations of u 1 , u 2 , , u l . Proof (for a simple case) w 1 = a 1 u 1 + a 2 u 2 , w 2 = b 1 u 1 + b 2 u 2 c 1 w 1 + c 2 w 2 = ( c 1 a 1 + c 2 b 1 ) u 1 + ( c 1 a 2 + c 2 b 2 ) u 2 The standard (basis) vectors of R n obviously, every vector in R n may be uniquely linearly combined by these standard vectors.
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Unformatted text preview: ( unique linear combination) What is the condition in R 3 ? in R n ? algebraically, this means that u and v are nonzero vectors, and u ≠ c v . 3 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = n n v v v A # " 2 1 2 1 ] [ a a a v If we write , then it is like a “dot product”. Example: then Property: A = and O v = for any A and v . Example: Property: Sometimes I n is simply written as I (any size). i th entry 4 Example: stochastic matrix probability matrix of a sample person’s residence movement nonnegative entries, unity column sums : current population of the city and suburbs : population distribution in the next year : population distribution in the year following the next Example: rotation matrix Proof for (e): If B ≠ A , then ( B − A ) e j ≠ , i.e., B e j ≠ A e j , for some j . 3...
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This note was uploaded on 10/16/2010 for the course EE [email protected] taught by Professor Fong during the Spring '09 term at National Taiwan University.

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102 - unique linear combination What is the condition in R...

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