Math 2331 Null Spaces, Column Spaces, & Linear Transformations

Math 2331 – Linear Algebra 4.2 Null Spaces, Column Spaces, & Linear Transformations Math 2331, Linear Algebra 1 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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 Math 2331, Linear Algebra 2 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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 = 0. Nul A = {x : x is in Rn and Ax = 0} (set notation) Theorem (2) The null space of an m × n matrix A is a subspace of Rn . Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn . Proof: Nul A is a subset of Rn since A has n columns. Must verify properties a, b and c of the definition of a subspace. Property (a) Show that 0 is in Nul A. Since , 0 is in . Math 2331, Linear Algebra 3 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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, and . Therefore 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) = c0 = 0. Since properties a, b and c hold, A is a subspace of Rn . Solving Ax = 0 yields an explicit description of Nul A. Math 2331, Linear Algebra 4 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Space: Example Example Find an explicit description of Nul A where   3 6 6 3 9 A= 6 12 13 0 3 Solution: Row reduce augmented matrix corresponding to Ax = 0:     3 6 6 3 9 0 1 2 0 13 33 0 ∼ ··· ∼ 6 12 13 0 3 0 0 0 1 −6 −15 0       x1 x2 x3 x4 x5     =        −2x2 − 13x4 − 33x5 x2 6x4 + 15x5 x4 x5 Math 2331, Linear Algebra       5 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Space: Example (cont.)    = x2    −2 1 0 0 0    +      x4    −13 0 6 1 0     +     x5    −33 0 15 0 1       Then Nul A =span{u, v, w} Math 2331, Linear Algebra 6 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Space: Observations Observations: 1. Spanning set of Nul A, found using the method in the last example, is automatically linearly independent:    c1    −2 1 0 0 0        + c2      −13 0 6 1 0        + c3      −33 0 15 0 1       =     0 0 0 0 0       =⇒ c1 = c2 = c3 = 2. If Nul A6= {0}, the the number of vectors in the spanning set for Nul A equals the number of free variables in Ax = 0. Math 2331, Linear Algebra 7 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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 = [a1 . . . an ], then Col A =Span{a1 , . . . , an } Theorem (3) The column space of an m × n matrix A is a subspace of Rm . Why? (Theorem 1, page 194) 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 Rn } Math 2331, Linear Algebra 8 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Column Space: Example Example Find a matrix A such that W = Col A where      x − 2y   3y : x, y in R . W =   x +y Solution:       x − 2y 1 −2  3y  = x  0  + y  3  x +y 1 1   =   x y Math 2331, Linear Algebra  9 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Column Space: Example (cont.) Therefore  A=  . By Theorem 4 (Chapter 1), The column space of an m × n matrix A is all of Rm if and only if the equation Ax = b has a solution for each b in Rm . Math 2331, Linear Algebra 10 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range The Contrast Between Nul A and Col A Example   1 2 3  2 4 7   Let A =   3 6 10  . 0 0 1 (a) The column space of A is a subspace of Rk where k = (b) The null space of A is a subspace of Rk where k = . . (c) Find a nonzero vector in Col A. (There are infinitely many possibilities.)   1  2   +  3  0   2  4   +  6  0    3  7       10  =  1 Math 2331, Linear Algebra     11 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range The Contrast Between Nul A and Col A (cont.) Example (cont.) (d) Find solution.  1  2   3 0 a nonzero vector in Nul A. Solve Ax = 0 and pick one  2 3 0 4 7 0   row reduces to 6 10 0  0 1 0 x1 = −2x2 x2 is free x3 = 0  1  0   0 0 2 0 0 0 0 1 0 0  0 0   0  0  =⇒ let x2 = =⇒   x1 x =  x2  =  x3   Contrast Between Nul A and Col A where A is m × n (see page 204) Math 2331, Linear Algebra 12 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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. Math 2331, Linear Algebra 13 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Spaces & Column Spaces: Review (cont.) Theorem (1, 2 and 3 in Sections 4.1 & 4.2) If v1 , . . . , vp are in a vector space V , then Span{v1 , . . . , vp } is a subspace of V . The null space of an m × n matrix A is a subspace of Rn . The column space of an m × n matrix A is a subspace of Rm . Math 2331, Linear Algebra 14 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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. Math 2331, Linear Algebra 15 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Spaces & Column Spaces: Examples (cont.) Example     x  x −y =0 (b) V =  y  : y +z =0   z Solution: Rewrite x −y =0 y +z =0 as      x   y  = 0 0 z   1 −1 0 So V =Nul A where A = . Since Nul A is a 0 1 1 subspace of R2 , V is a vector space. Math 2331, Linear Algebra 16 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Spaces & Column Spaces: Examples (cont.) Example    x +y   (c) S =  2x − 3y  : x, y , z are real   3y One Solution: Since       x +y 1 1  2x − 3y  = x  2  + y  −3  , 3y 0 3     1   1 S = span  2  ,  −3  ;   0 3 therefore S is a vector space by Theorem 1. Math 2331, Linear Algebra 17 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range Null Spaces & Column Spaces: Examples (cont.) Another Solution: Since       x +y 1 1  2x − 3y  = x  2  + y  −3  , 3y 0 3  S =Col A  1 1 where A =  2 −3 ; 0 3 therefore S is a vector space, since a column space is a vector space. Math 2331, Linear Algebra 18 / 19 4.2 Null Spaces, Column Spaces, & Linear Transformations Null Space Column Space Nul A & Col A Kernal and Range 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 1. 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) = 0. 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 . Math 2331, Linear Algebra 19 / 19