Linear Transformations, Null Spaces and Ranges

2.1 Linear Transformations Math 4377/6308 Advanced Linear Algebra 2.1 Linear Transformations, Null Spaces and Ranges Math 4377/6308, Advanced Linear Algebra Spring, 2015 1 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 2.1 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 Math 4377/6308, Advanced Linear Algebra Spring, 2015 2 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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) T (x + y ) = T (x) + T (y ) and (b) T (cx) = cT (x) 1 If T is linear, then T (0) = 0. 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 x1 , · · · , xn ∈ V and a1 , · · · , an ∈ F , T( n X i=1 ai x i ) = n X ai T (xi ). i=1 Math 4377/6308, Advanced Linear Algebra Spring, 2015 3 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Special Linear Transformations 1 The identity transformation IV : V → V : IV (x) = x, ∀x ∈ V . 2 The zero transformation T0 : V → W : T0 (x) = 0, ∀x ∈ V . Matrix Transformation Suppose A is m × n. The matrix transformation TA : Rn → Rm : TA (x) = Ax, ∀ ∈ Rn . Matrix A is an object acting on x by multiplication to produce a new vector Ax. Solving Ax = b amounts to finding all in Rn which are transformed into vector b in Rm through multiplication by A. Terminology Rn : domain of T Rm : codomain of T T (x) in Rm is the image of x under the transformation T Set of all images T (x) is the range of T Math 4377/6308, Advanced Linear Algebra Spring, 2015 4 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Matrix Transformations: Example Example  1 Let A =  2 0  Then if x =  0 1 . Define T : R2 −→ R3 by T (x) = Ax. 1      1 0   2 2 2 , T (x) = Ax =  2 1  = 5  1 1 0 1 1 Math 4377/6308, Advanced Linear Algebra Spring, 2015 5 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Matrix Transformations: Example Example  Let A =    2 1 −2 3 2 , u =  3 , b = and −5 10 −15 −10 1     3 c= . Define a transformation T : R3 → R2 by T (x) = Ax. 0 a. Find an x in R3 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: for x, or    x   1 −2 3  1  2 x2 = −5 10 −15 −10 x3 (a) Solve =  Math 4377/6308, Advanced Linear Algebra Spring, 2015 6 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Matrix Transformations: Example (cont.) Augmented matrix:     1 −2 3 2 1 −2 3 2 ∼ −5 10 −15 −10 0 0 0 0 x1 = 2x2 − 3x3 + 2 x2 is free x3 is free Let x2 = and x3 =  So x =  . Then x1 =  .  Math 4377/6308, Advanced Linear Algebra Spring, 2015 7 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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:     1 −2 3 3 1 −2 3 0 ∼ −5 10 −15 0 0 0 0 1 c is not in the of T . Math 4377/6308, Advanced Linear Algebra Spring, 2015 8 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Linear Transformations If A is m × n, then the transformation T (x) = Ax has the following properties: T (u + v) = A (u + v) = = + + and T (cu) = A (cu) = Au = T (u) for all u,v in Rn and all scalars c. Every matrix transformation is a linear transformation. Math 4377/6308, Advanced Linear Algebra Spring, 2015 9 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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) = 0: N(T ) = {x ∈ V : T (x) = 0}. 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 (2.1) 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 = {v1 , · · · , vn } is a basis for V , then R(T ) = span(T (β)) = span({T (v1 ), · · · , T (vn )}). Math 4377/6308, Advanced Linear Algebra Spring, 2015 10 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 = 0. Nul A = {x : x is in Rn and Ax = 0} (set notation) Theorem 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 4377/6308, Advanced Linear Algebra Spring, 2015 11 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 12 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra       Spring, 2015 13 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 14 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 15 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 = [a1 . . . an ], then Col A =Span{a1 , . . . , an } Theorem The column space of an m × n matrix A is a subspace of Rm . 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 Rn } Math 4377/6308, Advanced Linear Algebra Spring, 2015 16 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 17 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A Column Space: Example (cont.) Therefore  A=  . 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 4377/6308, Advanced Linear Algebra Spring, 2015 18 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra     Spring, 2015 19 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 Math 4377/6308, Advanced Linear Algebra Spring, 2015 20 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 21 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 22 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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. Math 4377/6308, Advanced Linear Algebra Spring, 2015 23 / 24 2.1 Linear Transformations Linear Transformations Nul A & Col A 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 4377/6308, Advanced Linear Algebra Spring, 2015 24 / 24