Understanding Linear Transformations in Real Analysis
In Real Analysis, linear transformations play a crucial role in vector spaces. This content delves into the definitions of linear combinations, independence, dimensions, bases, and theorems related to vector spaces. Explore how sets of vectors can span a space, the concept of bases, and the uniqueness of representations in vector spaces. This study aids in grasping the fundamental concepts essential for a deeper understanding of Real Analysis.
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Real Analysis Ms.A.Benazir Assistant Professor of Mathematics Department of Mathematics, Hajee Karutha Rowther Howdia College.
Linear transformation: Linear transformation: Definition: Definition: A non-empty set X R is a vector space, If x+y X x X x X, y X and for all scalar C If x1,x2, ..xk R and c1,c2, ck are scalars, Then the vectors c1x1 + c2x2 + . + ckxk is called a l linear combination inear combination of x1,x2, xk. If S R and if E is a set of all linear combination of elements of S then we say that S spans E (or) E is the span of S. Every span is a vector space vector space.
A set consisting of vectors x1,x2,..xk is said to be independent if the relation c1x1+c2x2+ +ckxk= 0 Implies that c1=c2= ..cn=0 otherwise x1,x2, ..xk is said to dependent. No independent set contains the null vector. If a vector space X contains an independent sets of r vector but contain no independent set of (r+1) vectors. We say that X has dimension r. That is, dim X = r
The set consisting of zero alone is a vector space is dimension is zero. An independent subset of a vector space X which span X is called a basis of X. If B = {x1,x2, ,xn} is a basis of X, Then ever x X has a unique representation of a form X = c1x1+c2x2+ .+crxr. r = i cixi That is, X= . 1
Such a representation exist since B spans X, and it is unique. Since B is independent the number, c1,c2, ..cr are called the co-ordinates of X with respect to the basis B. The set {e1,e2, .en} is the basis where ej is the vector in r whose j-th co-ordinate is 1 and all other co-ordinates are zero. If x r and x = (x1,x2, ..xn), then x = x1e1+x2e2+ .+xnen. We shall call e1,e2, en are standard basis.
Theorem : Theorem : Let r be a positive integer. If a vector space S is spanned by a set of r vector. Then, dim X r Proof: Proof: Let S0 = {x1,x2, x0} be the span of the vector space X. Suppose if the theorem is not true. That is, dim X > r It means that X has a set of {y1,y2, ..yr+1} of r+1 linearly independent vectors yi 0, i= 1,2, .r+1.
If y1 X and X is spanned by S0 That is y1 = 1x1 + 2x2 + ..+ rxr. Implies 1x1 = y1- 2x2 - - rxr. x1 = 1/ 1 [y1- 2x2 - - rxr] Since 1 0 Then we get, x1 = c1y1+c2y2+ ..+cryr Therefore the set S1 = {y1,x2, ..xr} be the open set of X for 0 i < r. Let Si = {y1,y2, ..yi,xi+1, ..xr} which spans X.
Then, yi+1=(1y1+2y2+..+iyi)+(i+1xi+1+.+xrr) Suppose i+1 = i+2 = = r = 0. Which is contradiction to the fact that y1,y2, yr+1 is linearly independent. Therefore i+1 = i+2 = = r 0. Assume that i+1 0. xi+1 = 1/ i+1 [yi+1- = j 1 r i jyj- jxj] = + 2 j i Therefore Si+1={y1,y2, ..yi+1,xi+2, ..xr} which spans X.
Similarly proceeding like this, We get Sr = {y1,y2, ..,yr} which spans X. Since yr+1 X, it can be written as, yr+1 = c1y1 + c2y2 + + cryr The set S is linearly dependent. Since yr+1 can be written as linear combination {y1,y2, ..yr}. Which is contradiction. {y1,y2, ..yr} is linearly independent is wrong. Therefore, X contains a set of r vectors and the dimension X r (dim X r ). Hence the theorem.
Corollary: Corollary: dim R = n Proof: Proof: Since {e1,e2, ..en} spans R by the above theorem. We have, {e1,e2, ..en} which implies dim R n (1) {e1,e2, ..en} is independent. Then we have dim R n (2) Hence proved.
Definition: Definition: A mapping A of a vector space X into a vector space Y is said to be a linear transformation, if A(x1+x2) = A(x1) + A(x2) and scalar multiplication. A(cx) = cA(x) for all x1,x2, ,x X and all scalars C. A(x) can be written as Ax If A is linear,A0=0. If A is linear, A of X into Y is completely determined by its action on any basis.
If x1 into xn is a basis of X then every x X has a unique representation of the form n x = cixi = i 1 and the linearity of A allow us to compute Ax from the vector Ax1,Ax2, Axn and the co-ordinates c1,c2, ,cn. By the formula, n Ax = ciAxi = i 1
Linear transformation of X into X are often called an linear operators on X. If A is a linear operators on X which, (i) one to one (ii) maps X onto X, we say that A is invertible We can define an operation A (Ax) = x for all x X Similarly A(A x) = x for all x X and that A is similar.
DEFinition DEFinition: : Let L(X,Y) be the set of all linear transformation of the vector space X into the vector space Y. If A1, A2 L(X,Y) and if C1,C2 are scalars. Define C1A1+C2A2 by, (C (C1 1A A1 1+C +C2 2A A2 2)x = C )x = C1 1A A1 1x + C where x X. x + C2 2A A2 2x x It is clear that C1A2+C2A2 L(X,Y).
Definition: Definition: If X,Y,Z are vector space if A L(X,Y), B L(X,Y,Z). We define this product BA by the composition of A and B. (BA)x = B(Ax) Therefore BA BA L(X,Z). BA need not be AB. Even if X=Y=Z
Definition: Definition: A differentiable mapping f of an open set E R into R is said to be continuously differentiable in E, If f is a continuous mapping of E into L(R ,R ). It is required that to every x E and to every > 0 corresponds a > 0 such that ||f (x)-f (y)|| < . If y E and |x-y| < . We also say that, F is a (E).
