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# Math 1554 - Linear Algebra Sections 1.3-1.5 MATH 1554 K3

Georgia Tech

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This 5 page Class Notes was uploaded by awesomenotes on Sunday February 14, 2016. The Class Notes belongs to MATH 1554 K3 at Georgia Institute of Technology taught by Stavros Garoufalidis in Spring 2016. Since its upload, it has received 66 views. For similar materials see Linear Algebra in Mathematics (M) at Georgia Institute of Technology.

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Date Created: 02/14/16

1.3 Vector Equations Vector – ordered lit of numbers − A matrix with only one column R – real numbers that appear as entries in the vectors Vectors in R 2 − R – (read “r-two) exponent 2 denotes that each vector contains two entries; the set of all 2ectors with two entries − two vectors in R are equal if and only if their corresponding entries are equal 4 7 − ≠ 7 4 2 − vectors in R are ordered pairs of real numbers Sum of Vectors (vector addition) 2 − given two vectors u and v in R , their sum in the vector u+v obtained by adding corresponding entries of u and v 2 1 + 2 3 − 1 + = = −2 5 −2 + 5 3 Scalar multiple − given a vector u and a real number c, the scalar multiple of u by c is cu obtained by multiplying each entry in u by c 3 3 15 − ???? = −1 ,???? = 5,then ???????? = 5−1 = −5 − scalar – a (real) nu2ber used to multiple a vector or matric Geometric Description of R a − geometric point (a,b) = column vectorb Parallelogram Rule for Addition 2 − if u and v in R are represented as points in the plane, then u+v corresponds to the forth vertex of the parallelogram whose other vertices are u, 0, v 3 Vector in R − 3x1 column matrices with three entries − represented geometrically by points in 3D coordinate space, with arrows from the origin sometimes included for visual clarity n Vectors in R − R – (read “r-n”) denoted the collection of all lists (or ordered n-tuples) of n real numbers, usually written in nx1 column matrices such as u1 ???? = … un − Zero vector – vector whose entries are all zero; denoted by 0 (number of entries in 0 will be cleared from context) − Equality of vectors in R and operations of scalar multiplication and vector n 2 addition in R are defined entry by entry as in R n n Algebraic Properties of R : (for all u, v, wand scalars c,d) 1. ???? + ???? = ???? + ???? 5. ???? ???? + ???? = ???????? + ???????? 2. (???? + ????) + ???? = ???? + (???? + ????) 1. ???? + ???? ???? = ???????? + ???????? 3. ???? + ???? = ???? + ???? = ???? 2. ???? ???????? = ???????? ???? 4. ???? + −???? = −???? + ???? = ???? 3. ???????? = ???? Linear Combinations − a sum of scalar multiples of vectors − scalars are called weights − given vectors v , v , …, v in R and given scalars c , c , …, c , the vector y 1 2 p 1 2 p (linear combination) is defined by y = ????▯ ▯+ ⋯+ ???? ????▯ ▯ − weights in linear combinations can be any real numbers including 0 A vector equation ???? ▯ ▯ ???? ????▯ ▯⋯+ ???? ???? = ▯ ▯as the same solution set as the linear system whose augmented matrix is ???? ▯ ????▯ … ???? ▯ In particular, b can be generated by a linear combination of a 1…,a nf and only if there exists a solution to the linear system corresponding to the matrix If v ,…, v are in R , then the set of all linear combinations of v ,…, v is denoted by 1 p n 1 p Span{ v 1…, vp} and is called the subset of R spanned (or generated) by v ,…,1v . p That is, Span{ v1,…, p } is the collection of all vectors that can be written in the form ????▯ ▯+ ⋯+ ???? ????▯ ▯th c ,…1 c scplars − asking whether a vector b is in Span{ v ,…, v } = asking whether the vector 1 p equation ????▯ ▯+ ???? ????▯ ▯⋯+ ???? ???? =▯ ▯has a solution A Geometric Description of Span{v} and Span{u,v} 3 − let v be a nonzero vector in R – Span{v} is the set of all scalar multiples of v, 3 which is the set of points on the link in R through v and 0 3 − if u and v are nonzero vectors in R , with v not a multiple of u, then Span{u,v} is the plane in R that contains u,v, and 0. In particular, Span{u,v} contains the 3 line in R through u and 0 and the line through v and 0. 1.4 The Matrix Equation Ax=b − view linear combination of vectors ad the products of a matrix and a vector 2 − If A is an mxn matrix, with columns1a ,n,a and if x is in R , then the product of A and x, denoted by Ax, is the linear combination of the columns of A using ????▯ the corresponding entries in x as weights; that is A▯ =▯ ???? ,▯ ,…,???? = ???? ▯ ▯ ▯ + ????▯ ▯+ ⋯+ ???? ▯ ▯ o Note: Ax is only defined is the number of columns of A equals the number of entries in x − Example 1 2 −1 4 1 2 −1 4 6 −7 3 3 = 4 + 3 + 7 = + + = 0 −5 3 7 0 −5 3 0 −15 21 6 Theorem 3 m If A is anxm n matrix, with colum1s an,…,a , and if b is in R , the matric equation Ax=b has the same solution set as the vector equation ????▯ ▯+ ???? ▯ ▯ ⋯+ ???? ????▯ ▯b which, in turn, has the same solution set as the system of linear equations whose augmented matrix is ????▯ ????▯ … ▯ ???? Existence of Solutions − The equation Ax=b has a solution if and only if b is a linear combination of the columns of A Theorem 4 Let A be an augmented matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false m 1. For each b in R , the equation Ax=b has a solution m 2. Each b in R is a linearmcombination of the columns of A 3. The columns of A span R 4. A has a pivot position in every row Warning: Theorem 4 is about a coefficient matrix, not an augmented matrix. If an augmented matrix has a pivot position in every row, the equation Ax=b may or may not be consistent. Computation of Ax − Row-Vector Rule for Computing Ax − If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries rom row I of A and from the vector x − Identity matrix – a matrix with 1’s on the diagonal and 0’s elsewhere; denoted by I n − Ix=x for every x in R Properties of the Matrix- Vector Product Ax Theorem 5 n If A is an x n matrix, u and v are vectors in R and c is a scalar, then: 1. ???? ???? + ???? = ???????? + ???????? 2. ???? ???????? = ????(????????) 1.5 Solution Sets of Linear Systems Homogeneous Linear Systems − Homogeneous equations – can be written in the form Ax=0. Where A is an m m x matrix and 0 is the zero vector in R ; always has at least one solution, namely, x=0, the trivial solution − Trivial solution- zero vector that satisfies Ax=0 − Nontrivial solution – a nonzero vector x that satisfies Ax=0 − The homogeneous equation Ax=0 has a nontrivial solution if and only if the equation has at least one free variable Parametric Vector Form − Whenever a solution set is described explicitly with vectors − Implicit vs explicit equation definitions − Parametric vector equation − ???? = ???????? + ???????? Solutions of Nonhomogeneous Systems − think of vector addition as a translation 2 3 − given p and v in R or R , the effect of adding p to v is to move v in a direction parallel to the line through p and 0 − v is translated by p to v+p − the equation of the line through p parallel to v − x = p + tv − the solution set of Ax=b is a line through p parallel to the solution set of Ax=0 Theorem 6 Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form ???? = ???????? ???? where v hs any solution of the homogeneous equation Ax=0 Writing A Solution Set (of a consistent system) In Parametric Vector Form: 1. Row reduce the augmented matrix to reduced echelon form 2. Express each basic variable in terms of any free variables appearing in an equation 3. Write a typical solution x as a vector whose entries depend on the free variables, if any 4. Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters

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