Math 4377/6308 Advanced Linear Algebra

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1 2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston math.uh.edu/ jiwenhe/math4377 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 / 24

2 2. Linear Transformations, Null Spaces and Ranges Linear Transformations: Definition and Properties Matrix Transformations Matrix Acting on Vector Matrix-Vector Multiplication Transformation: Domain and Range Examples Null Spaces and Ranges Definition and Theorems Null Spaces and Column Spaces of Matrices Nullity and Rank Definitions and Dimension Theorem Ranks of Matrices and Rank Theorem Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 2 / 24

3 Linear Transformations Definition We call a function T : V W a linear transformation from V to W if, for all x, y V and c F, we have (a) (b) T (x + y) = T (x) + T (y) and T (cx) = ct (x) If T is linear, then T () =. 2 T is linear T (cx + y) = ct (x) + T (y), x, y V, c F. 3 If T is linear, then T (x y) = T (x) T (y), x, y V. 4 T is linear for x,, x n V and a,, a n F, n T ( a i x i ) = i= n a i T (x i ). i= Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 3 / 24

4 Special Linear Transformations The identity transformation I V : V V : I V (x) = x, x V. 2 The zero transformation T : V W : T (x) =, x V. Matrix Transformation Suppose A is m n. The matrix transformation T A : R n R m : T A (x) = Ax, R n. Matrix A is an object acting on x by multiplication to produce a new vector Ax. Solving Ax = b amounts to finding all in R n which are transformed into vector b in R m through multiplication by A. Terminology R n : domain of T R m : codomain of T T (x) in R m is the image of x under the transformation T Set of all images T (x) is the range of T Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 4 / 24

5 Matrix Transformations: Example Example Let A = 2 [ 2 Then if x =. Define T : R 2 R 3 by T (x) = Ax., T (x) = Ax = 2 [ 2 = 2 5 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 5 / 24

6 Matrix Transformations: Example Example [ Let A = [ 3 c = , u = 2 3 [, b = 2 and. Define a transformation T : R 3 R 2 by T (x) = Ax. a. Find an x in R 3 whose image under T is b. b. Is there more than one x under T whose image is b. (uniqueness problem) c. Determine if c is in the range of the transformation T. (existence problem) Solution: (a) Solve = for x, or [ x 2 3 [ x = x 3 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 6 / 24

7 Matrix Transformations: Example (cont.) Augmented matrix: [ x = 2x 2 3x x 2 is free x 3 is free [ Let x 2 = and x 3 =. Then x =. So x = Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 7 / 24

8 Matrix Transformations: Example (cont.) (b) Is there an x for which T (x) = b? Free variables exist There is more than one x for which T (x) = b (c) Is there an x for which T (x) = c? This is another way of asking if Ax = c is. Augmented matrix: [ [ 2 3 c is not in the of T. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 8 / 24

9 Linear Transformations If A is m n, then the transformation T (x) = Ax has the following properties: and T (u + v) = A (u + v) = + = + T (cu) = A (cu) = Au = T (u) for all u,v in R n and all scalars c. Every matrix transformation is a linear transformation. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 9 / 24

10 Null Space and Range Definition For linear T : V W, the null space (or kernel) N(T ) of T is the set of all x V such that T (x) = : N(T ) = {x V : T (x) = }. The range (or image) R(T ) of T is the subset of W consisting of all images of vectors in V : R(T ) = {T (x) : x V }. Theorem (2.) For vector spaces V, W and linear T : V W, N(T ) and R(T ) are subspaces of V and W, respectively. Theorem (2.2) For vector spaces V, W and linear T : V W, if β = {v,, v n } is a basis for V, then R(T ) = span(t (β)) = span({t (v ),, T (v n )}). Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 / 24

11 Null Space of a Matrix The null space of an m n matrix A, written as Nul A, is the set of all solutions to the homogeneous equation Ax =. Nul A = {x : x is in R n and Ax = } (set notation) Theorem The null space of an m n matrix A is a subspace of R n. Equivalently, the set of all solutions to a system Ax = of m homogeneous linear equations in n unknowns is a subspace of R n. Proof: Nul A is a subset of R n since A has n columns. Must verify properties a, b and c of the definition of a subspace. Property (a) Show that is in Nul A. Since, is in. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 / 24

12 Null Space (cont.) Property (b) If u and v are in Nul A, show that u + v is in Nul A. Since u and v are in Nul A, Therefore and. A (u + v) = + = + =. Property (c) If u is in Nul A and c is a scalar, show that cu in Nul A: A (cu) = A (u) = c =. Since properties a, b and c hold, A is a subspace of R n. Solving Ax = yields an explicit description of Nul A. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 2 / 24

13 Null Space: Example Example Find an explicit description of Nul A where [ A = Solution: Row reduce augmented matrix corresponding to Ax = : [ x x 2 x 3 x 4 x 5 = [ x 2 3x 4 33x 5 x 2 6x 4 + 5x 5 x 4 x 5 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 3 / 24

14 Null Space: Example (cont.) Then = x x x 5 Nul A =span{u, v, w} 33 5 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 4 / 24

15 Null Space: Observations Observations:. Spanning set of Nul A, found using the method in the last example, is automatically linearly independent: = c 2 + c c c = c 2 = c 3 = = 2. If Nul A {}, the the number of vectors in the spanning set for Nul A equals the number of free variables in Ax =. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 5 / 24

16 Column Space of a Matrix The column space of an m n matrix A (Col A) is the set of all linear combinations of the columns of A. If A = [a... a n, then Col A =Span{a,..., a n } Theorem The column space of an m n matrix A is a subspace of R m. Why? Recall that if Ax = b, then b is a linear combination of the columns of A. Therefore Col A = {b : b =Ax for some x in R n } Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 6 / 24

17 Column Space: Example Example Find a matrix A such that W = Col A where x 2y W = 3y : x, y in R. x + y Solution: x 2y 3y x + y = = x + y [ x y 2 3 Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 7 / 24

18 Column Space: Example (cont.) Therefore A =. The column space of an m n matrix A is all of R m if and only if the equation Ax = b has a solution for each b in R m. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 8 / 24

19 The Contrast Between Nul A and Col A Example Let A = (a) The column space of A is a subspace of R k where k =. (b) The null space of A is a subspace of R k where k =. (c) Find a nonzero vector in Col A. possibilities.) (There are infinitely many 3 7 = Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 9 / 24

20 The Contrast Between Nul A and Col A (cont.) Example (cont.) (d) Find a nonzero vector in Nul A. solution row reduces to Solve Ax = and pick one 2 x = 2x 2 x 2 is free x 3 = = let x 2 = = x = x x 2 x 3 = Contrast Between Nul A and Col A where A is m n Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 2 / 24

21 Null Spaces & Column Spaces: Examples Example Determine whether each of the following sets is a vector space or provide a counterexample. {[ } x (a) H = : x y = 4 y Solution: Since = is not in H, H is not a vector space. Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, 25 2 / 24

22 Null Spaces & Column Spaces: Examples (cont.) Example (b) V = x y z : x y = y + z = Solution: Rewrite x y = as y + z = x y z [ So V =Nul A where A = subspace of R 2, V is a vector space. = [. Since Nul A is a Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, / 24

23 Null Spaces & Column Spaces: Examples (cont.) Example (c) S = x + y 2x 3y 3y : x, y, z are real One Solution: Since x + y 2x 3y 3y = x 2 + y 3 3, S = span therefore S is a vector space. 2, 3 3 ; Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, / 24

24 Null Spaces & Column Spaces: Examples (cont.) Another Solution: Since x + y 2x 3y = x 3y 2 + y S =Col A where A = therefore S is a vector space, since a column space is a vector space., ; Jiwen He, University of Houston Math 4377/638, Advanced Linear Algebra Spring, / 24