Math 2331 Linear Algebra

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1 4.2 Null Spaces, Column Spaces, & Linear Transformations Math 233 Linear Algebra 4.2 Null Spaces, Column Spaces, & Linear Transformations Jiwen He Department of Mathematics, University of Houston math.uh.edu/ jiwenhe/math233 Jiwen He, University of Houston Math 233, Linear Algebra / 9

2 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Spaces Definition Theorem Examples Column Spaces Definition Theorem Examples The Contrast Between Nul A and Col A Null Spaces & Column Spaces: Review Null Spaces & Column Spaces: Examples Kernal and Range of a Linear Transformation Jiwen He, University of Houston Math 233, Linear Algebra 2 / 9

3 Null Space Null Space 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 (2) 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 233, Linear Algebra 3 / 9

4 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 233, Linear Algebra 4 / 9

5 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 233, Linear Algebra 5 / 9

6 Null Space: Example (cont.) Then = x x x 5 Nul A =span{u, v, w} 33 5 Jiwen He, University of Houston Math 233, Linear Algebra 6 / 9

7 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 233, Linear Algebra 7 / 9

8 Column Space Column Space 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 Theorem (3) Col A =Span{a,..., a n } The column space of an m n matrix A is a subspace of R m. Why? (Theorem, page 94) 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 233, Linear Algebra 8 / 9

9 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 233, Linear Algebra 9 / 9

10 Column Space: Example (cont.) Therefore A =. By Theorem 4 (Chapter ), 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 233, Linear Algebra / 9

11 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 233, Linear Algebra / 9

12 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 (see page 24) Jiwen He, University of Houston Math 233, Linear Algebra 2 / 9

13 Null Spaces & Column Spaces: Review Review A subspace of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H. b. For each u and v in H, u + v is in H. (In this case we say H is closed under vector addition.) c. For each u in H and each scalar c, cu is in H. (In this case we say H is closed under scalar multiplication.) If the subset H satisfies these three properties, then H itself is a vector space. Jiwen He, University of Houston Math 233, Linear Algebra 3 / 9

14 Null Spaces & Column Spaces: Review (cont.) Theorem (, 2 and 3 in Sections 4. & 4.2) If v,..., v p are in a vector space V, then Span{v,..., v p } is a subspace of V. The null space of an m n matrix A is a subspace of R n. The column space of an m n matrix A is a subspace of R m. Jiwen He, University of Houston Math 233, Linear Algebra 4 / 9

15 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 233, Linear Algebra 5 / 9

16 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 233, Linear Algebra 6 / 9

17 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 2, 3 3 therefore S is a vector space by Theorem. ; Jiwen He, University of Houston Math 233, Linear Algebra 7 / 9

18 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 233, Linear Algebra 8 / 9

19 Kernal and Range of a Linear Transformation Linear Transformation A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T (x) in W, such that. T (u + v) = T (u) +T (v) for all u, v in V ; 2. T (cu) =ct (u) for all u in V and all scalars c. Kernal and Range The kernel (or null space) of T is the set of all vectors u in V such that T (u) =. The range of T is the set of all vectors in W of the form T (u) where u is in V. So if T (x) =Ax, col A =range of T. Jiwen He, University of Houston Math 233, Linear Algebra 9 / 9