University:
University of California, Berkeley
Course:
MATH 56 | Linear Algebra
Academic year:

2022
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383

Pages:

2
Author:

CometConductor

Linear Transformation Matrix Definition: The n ull space of an m x n matrix A, written as Nul A, is the set of all solutions of → − → − the homogeneous equation A x = 0 . In set notation, → − − − − Nul A = { → x: → x is in Rn and A→ x = 0 }. 1 3 Exercise 1. For A = 0 1 span the null space. (#3) 5 0 , find an explicit description of Nul A by listing vectors that 4 −2  1 −2 0 4 Class Exercise 1. For A = 0 0 1 −9 0 0 0 0 vectors that span the null space. (#5)  0 0, find an explicit description of Nul A by listing 1 Definition: The column space of an m x n matrix A, written as Col A, is the set of all linear − − combinations of the columns of A. If A = [ → a 1 ··· → a n ], the − − Col A = Span { → a 1, . . . . . . , → a n }. → − − x = b Fact: The column space of an m x n matrix A is all of Rm if and only if the equation A→ → − has a solution for each b in Rm . Exercise 2. Consider the matrix   1 −1 A = 0 1 . 3 −3   1 → − Determine whether b = 2 is in the column space of A. (Poole Section 3.5 Example 3.41) 3 −6 Class Exercise 2. Let A = −3 12 2 → − − and w = . Determine if → w is in Col A. (#23) 6 1 Exercise 3. Either use an appropriate theorem to show that the set   a {  b : a + b + c = 2 } c is a vector space, or find a specific example to the contrary. (#7) Exercise 4. Either use an appropriate theorem to show that the set   a b  {  c : a − 2b = 4c, 2a = c + 3d } d is a vector space, or find a specific example to the contrary. (#9) Class Exercise 3. Either use an appropriate theorem to show that the set   b − 2d 5+d  { b + 3d: b, d real } d is a vector space, or find a specific example to the contrary. (#11) Class Exercise 4. Either use an appropriate theorem to show that the set   c − 6d {  d : c, d real } c is a vector space, or find a specific example to the contrary. (#13) Definition: 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 → − → − → − → − → − → − (i) T ( u + v ) = T ( u ) + T ( v ) for all u and v in V , and − − − (ii) T (c→ u ) = cT (→ u ) for all → u in V and all scalars c. Theorem: The null space of an m x n matrix A is a subspace of Rn . Equivalently, the set → − − of all solutions to a system A→ x = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn . Theorem: The column space of an m x n matrix A is a subspace of Rm .   2 −6 −1 3  k  Exercise 5. For the matrix A =  −4 12 , (a) find k such that Nul A is a subspace of R and 3 −9 (b) find k such that Col A is a subspace of Rk . (#17) 4 5 −2 6 0 Class Exercise 5. For the matrix A = , (a) find k such that Nul A is a 1 1 0 1 0 k subspace of R and (b) find k such that Col A is a subspace of Rk . (#19)

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