Linear Algebra (I) Yijia Chen. linear transformations and their algebraic properties. 1. A Starting Point. y := 3x.

Transcription

1 Linear Algebra I) Yijia Chen Linear algebra studies Exaple.. Consider the function This is a linear function f : R R. linear transforations and their algebraic properties.. A Starting Point y := 3x. Geoetrically it defines a line on the two-diensional plane/space R. Algebraically, for all x, x, s, x R fx + x ) = x + x fsx) = s fx). Exaple. Newton s second law). F = a or equivalently a = F. That is, a force F applied to an object of ass causes an acceleration of a. In two diensional space, a force F is a vector F, F ). Then F a = a, a ) =, F ). This transforation is still linear: F + F, F + F ) sf, sf ) = F = s, F ) F, F F +, F We will define addition and ultiplication for vectors ore precisely later. If we are in three diensions, then Then ). F := F x, F y, F z ) and a := a x, a y, a z ) ) a x := a x + 0 a y + 0 a z a y := 0 a x + a y + 0 a z a z := 0 a y + 0 a y + a z.

2 This linear transforation is associated with the atrix Algebra: anipulate sybolic cobinations of objects and equate one such cobination with another. E.g., how to siplify an expression such as x 3)x + 5) = x + x 5. In linear algebra we shall anipulate not just scalars, but also vectors, vector spaces, atrices, and linear transforations. - Analysis: estiate and approxiating objects to obtain qualitative rather than quantitative properties. - Cobinatorics: count objects and equate the.. Vectors and Vector Spaces Scalars. Roughly, a scalar is any quantity which can be described by a single nuber, and for all our purposes, a nuber in R. For instance, in physics we have ass, speed, length, tie and acceleration, and in finance, oney, interest rates, prices, and volue. The set of all scalars is referred to as the field of scalars; it is usually just R, the field of real nubers. Any two scalars can be added, subtracted, or ultiplied together to for another scalar. Scalars obey various rules of algebra, for instance x + y = y + x x y + z) = x y + x z. Vectors and vector spaces. Roughly, a vector is any eber of a vector space; a vector space is any class of objects which can be added together, or ultiplied with scalars. As with scalars, vectors ust obey certain rules of algebra. Exaple.. The two diensional is a vector space with More generally, the n diensional has R = { x, y) x, y R } x, y ) + x, y ) = x + x, y + y ) cx, y) = cx, cy). R n = { x,..., x n ) x,..., x n R } x,..., x n ) + y,..., y n ) = x + y,..., y n + y n ) cx,..., x n ) = cx,..., cx n ). Definition.. A vector space V is a collection of objects called vectors) for which two operations can be perfored:

3 - Vector addition: Let u, v V. Then u + v W. That is, addition is a function + : V V V. - Scalar ultiplication: Let c R and v V. Then cv = c v V. That is, ultiplication is a function : R V V Moreover, they satisfy the following algebraic laws. I) Addition is coutative: u + v = v + u. II) Addition is associative: u + v + w) = u + v) + w. III) Addition has an identity: There is a vector 0 V, i.e., the zero vector, such that 0 + v = v. IV) Addition has inverse: For every v there is a vector u V such that u + v = 0. That is, u is the additive inverse of v, also denoted by v. V) Multiplication has an identity, i.e., v = v for every v V. VI) Multiplication is associative: For all a, b R and v V abv) = ab)v. VII) ultiplication is linear: For all a R and u, v V we have au + v) = au + av. VIII) Multiplication distributes over addition: For all a, b R and v V a + b)v = av + bv. Now we can verify forally R n defined in Exaple. is a vector space. Aong others, ) The zero vector is 0,..., 0); ) the additive inverse of x,..., x n ) is x,..., x n ). Matrices as vectors. Let, n. An n atrix A has the for a a a n a a a n a a a n Soeties, we write A M n R), where M n R) is the vector space defined by a a a n b b b n a + b a + b a n + b n a a a n... + b b b n... = a + b a + b a n + b n... a a a n b b b n a + b a + b a n + b n and a a a n ca ca ca n a a a n c... = ca ca ca n a a a n ca ca ca n 3

4 All the laws I) VIII) again can be verified easily, in particular the zero atrix is.... Derived algebraic laws. Lea.3 Vector cancellation law). If u + v = u + w, then v = w. Proof: We can deduce u + v = u + w by assuption = u) + u + v) = u) + u + w) = u + u) + v = u + u) + w by II) = 0 + v = 0 + w by IV) = v = w by III). Siilarly we can prove: Lea.4. ) 0v = 0. ) )v = v. 3) v + w) = v) + w). 4) a0 = 0. 5) a x) = a)x = ax Recall an n atrix has the for 3. Back to the Textbook a a a n a a a n a a a n We let entrya, i, j) = a ij. The transpose atrix of A, denoted by A T, is the n atrix a a a a a a a n a n a n In other words, entrya T, i, j) = entrya, j, i). 4

5 If n = and A = A T, then A is a syetric atrix. A special syetric atrix is , 0 0 i.e., the identity atrix, denoted by I or I n. Triangular atrices. Let A be an n n atrix, if entrya, i, j) = 0 for all i > j, then A is an upper triangular atrix. Syetrically, if entrya, i, j) = 0 for all i < j, then A is a lower triangular atrix. If an upper lower) triangular atrix A satisfies entrya, i, i) = 0 for all i, then A is a strictly upper lower, respectively) triangular atrix. 3.. Matrix ultiplication. Let A M,l R) and B M l,n R). Then the product AB is an n atrix defined by entryab, i, j) = l entrya, i, k) entryb, k, j) k= = a i b j + a i b j + + a il b lj. 5