Rank Nullity Theorem

3 Key excerpts on "Rank Nullity Theorem"

• 

  eBook - ePub

  Linear Algebra

  • Lina Oliveira(Author)
  • 2022(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Hence, N (T) = N ([ T ]) = { (0, 0) }. Maybe it is not so easy to justify that I (T) = ℝ 2. The next theorem will help. Let T be a linear transformation. The nullity of T, nul (T), is the dimension of its null space and the rank of T, rank (T), is the dimension of its image. Theorem 5.2 (Rank-nullity theorem). Let T : K n → K k be a linear transformation. Then n = nul (T) + rank (T). Before proving the result, we apply it to c) in Example 5.6. We have that 2 = dim N (T) + dim I (T) = 0 + dim I (T), yielding that I (T) is a subspace having dimension 2 within ℝ 2. Hence, the only possibility is I (T) = ℝ 2, that is, T is surjective. Proof Let A be the k × n matrix of T relative to the bases E n and E k. Then, by Proposition 3.11 and Theorem 3.6, n = dim N (A) + rank (A) = dim N (A) + dim C (A) = dim N (T) + dim I (T), which concludes the proof. 5.3.2 Linear transformations T : U → V Let U and V be vector spaces over K, let B 1 = (u 1, u 2, …, u n) be a basis of U and. let B 2 = (v 1, v 2, …, v k) be a basis of V. Let T : U → V be a linear transformation. We are now interested in devising a way to determine the null space and the image of such a general linear transformation by means of its representing matrix relative to the bases of the domain and the codomain, as we did in § 5.3.1, for the particular kind of linear transformations under scrutiny in that part of the book. Tackling firstly the null space of T : U → V, we are then interested in determining the vectors x ∈ U such that T (x) = 0. If A = [ T ] B 2, B 1 is the matrix of T relative to the bases of the domain and codomain, we have T (x) = 0 if and only if [ T (x) ] B 2 = 0. It follows that T (x) = 0 if and only if A [ x ] B 1 = 0, where this equality corresponds to determining the null space of A

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  eBook - ePub

  Linear Algebra with Applications

  • Roger Baker, Kenneth Kuttler(Authors)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  It follows at once that for an m × n matrix A,

  N(A) + R(A) = n.

  Also, it is clear at this point that for A an m × n matrix, N (A) equals the number of non pivot columns. This is because it has already been observed that R (A) is the number of pivot columns. Thus it is very easy to compute the rank and nullity of a matrix.

  Example 5.15. Here is a matrix.

  Find its rank and nullity. Using Maple or simply doing it by hand, the row reduced echelon form is Therefore, there are three pivot columns and two nonpivot columns, so the rank is 3 and the nullity is 2.

  5.5 Rank and nullity of a product

  We note some useful results of Sylvester.

  Lemma 5.3. Let T : V → W and U : W → Z be linear mappings, with finite dimensional spaces V, W.

  (i) R(UT) ≤ min{R(U), R(T)}.

  (ii) If T is a bijection, then

  R(UT) = R(U).

  (iii) If U is a bijection, then

  R(UT) = R(T).

  Proof. (i) Clearly

  Im UT ⊆ Im U.

  So Im UT has dimension no more than the dimension of Im U and

  R(UT) ≤ R(U).

  Let υ1 , …, υm be the vectors constructed in the proof of Theorem 5.1 . Clearly

  (ii) If T is a bijection then

  so of course R(UT) = R(U).

  (iii) Let υ1 , …, υm be as above. If U is a bijection, then

  UTυ

  k + 1

  , ···, UTυm

  is a linearly independent set. For an equation

  x

  k + 1

  UTυ

  k + 1

  + ··· + xm UT υm = 0

  leads to an equation

  x

  k + 1

  Tυ

  k+ 1

  + ··· + xm Tυm = 0

  and thence to x

  k + 1

  = ··· = xm = 0. Now (5.6) gives

  R(UT) = m − k = R(T).

  Lemma 5.4. Let T : V → W where ker T is finite-dimensional, and let Z be a finite-dimensional subspace of W. Then

  VZ = {υ ∈ V : Tυ ∈ Z}

  is a subspace of V having dimension ≤ dim Z + N(T).

  Proof. It is easy to check that VZ is a subspace of V using the closure property of Lemma 4.3 . We ‘restrict’ T to VZ , that is consider the mapping T′ : V

  Z

  → Z defined by T′υ = Tυ (υ ∈ VZ

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  Linear Algebra

  • L. Shen, Haohao Wang, J. Wojdylo(Authors)
  • 2019(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  0}.

  Theorem 2.1.1

  Let T be a linear transformation from V into W . Then T is non-singular if and only if T carries each linearly independent subset of V onto a linearly independent subset of W.

  Proof : Let T : V → W be a linear transformation and non-singular, and v1 , …, vn linearly independent. To show T (v1 ), …, T (vn ) are linearly independent, consider

  Thus, T (v1 ), … T (vn ) are linearly independent.

  To show T is non-singular, we only need to show that T (v) = 0 implies v = 0. To do so, let v1 , … vn be a basis of V , and v = c 1 v1 + … +

  cn

  vn , then

  Thus T is non-singular.

  EXAMPLE 2.1.3

  In 2-dimensional space 2 linear maps are described by 2 × 2 real matrices relative to the standard basis. These are some examples:

  1.

  Rotation by 90 degrees counterclockwise:

  2.

  Rotation by angle θ counterclockwise:

  3.

  Reflection against the x -axis:

  4.

  Reflection against the y -axis:

  5.

  Scaling by k > 0 in all directions:

  6.

  Horizontal shear mapping:

  7.

  Vertical squeeze mapping, k > 1:

  8.

  Projection onto the y -axis:

  2.2Rank a nd Nullity of a Linear Transformation

  Definition 2.2.1

  Let V and W be vector spaces over the field , and let T be a linear transformation from V into W. The null space of T is the set of all vectors v ∈ V such that T (v) = 0. If V is finite-dimensional, the rank of T is the dimension of the range of T , and the nullity of T is the dimension of the null space of T.

  EXAMPLE 2.2.1

  Let [s] denote the ring of polynomials in a single variable s with coefficients from the field . A polynomial vector v ∈ ( [s ])w is a vector of size w with each entry being a polynomial. The degree n of a vector v ∈ ( [s ])w is the maximum amongst the degrees of its polynomial components. Alternatively we can write v as a polynomial of degree n with the coefficients being the vectors from w . Hence, ( [s])w = w [s ]. Similarly a polynomial matrix ∈ g ×w [s ] is a matrix of size g × w with the entries from [s ]. The degree of a polynomial matrix is the maximum of the degrees amongst its polynomial entries. A polynomial matrix can be written as a polynomial in s with coefficients being the matrices from g ×w. The null space of is {v ∈ w [s ] | R v = 0

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