What is Null space: Definition and 72 Discussions

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically:












{\displaystyle \ker(L)=\left\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \right\}.}

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  1. mido hoss

    Optimizing Null Space Solutions: How to Remove Zeros in Rank Deficient Matrices

    During calculating null space from rref matrix some rows are with 1 variable so it give this variable value of zero so the null space of rank deficient matrix be for example {-1,2,0,3,-4,0} my question is how to get rid of zeros in the null space solution and only solve it as basic and free...
  2. karush

    MHB Matrices.......whose null space consists all linear combinations

    $ v=\left[\begin{array}{r} -3\\-4\\-5\\4\\-1 \end{array}\right] w=\left[\begin{array}{r} -2\\0 \\1 \\4 \\-1 \end{array}\right] x=\left[\begin{array}{r} 2\\3 \\4 \\-5 \\0 \end{array}\right] y=\left[\begin{array}{r} -2\\1 \\0 \\-2 \\7 \end{array}\right] z=\left[\begin{array}{r} -1\\0 \\2 \\-3...
  3. archaic

    Finding a matrix from a given null space

    I have solved the exercise, so I'm not giving the vectors explicitly. I just want to know if there is a quicker way than mine. We know that ##A## must have ##4## columns and ##4## lines, and we also know that its nullity is ##2##, thus its rank is ##2##. I took the simplest matrix that can have...
  4. N

    I 2 and 3 dimensional invariant subspaces of R4

    I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4. I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
  5. karush

    MHB Null Space of A: Find Rank & Dim.

    Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...
  6. sarumman

    Proving or Disproving Null Space Containment in F(n) for A and A^2

    Homework Statement given I am required to proove or disprove:[/B] Homework Equations rank dim null space The Attempt at a Solution I tried to base my answer based on the fact that null A and null A^2 is Contained in F (n) and dim N(A)+rank(A)=N same goes for A^2.
  7. F

    Matrices:- Range and null space

    Homework Statement Question is uploaded I have completed till part iii and obtained correct answers i. 2 ii. Basis for R:- { ( 2 3 -1 ) , (1 4 2 ) } Cartesian equation; 2x-y+z=0 iii. Basis for Null:- { ( -3 2 0 1 ) , (2 -3 1 0 ) } 2. The attempt at a solution I have problem in last part. I...
  8. Zero2Infinity

    Write a matrix given the null space

    Homework Statement Build the matrix A associated with a linear transformation ƒ:ℝ3→ℝ3 that has the line x-4y=z=0 as its kernel. Homework Equations I don't see any relevant equation to be specified here . The Attempt at a Solution First of all, I tried to find a basis for the null space by...
  9. TheSodesa

    A real parameter guaranteeing subspace invariance

    Homework Statement Let ##A## and ##B## be square matrices, such that ##AB = \alpha BA##. Investigate, with which value of ##\alpha \in \mathbb{R}## the subspace ##N(B)## is ##A##-invariant. Homework Equations If ##S## is a subspace and ##A \in \mathbb{C}^{n \times n}##, we define multiplying...
  10. S

    MHB Question for null space of a matrix

    Let A be a 4×3 matrix and let c=2a1+a2+a3 (a) If N(A) = {0}, what can you conclude about the solutions to the linear system Ax=c? (b) If N(A) ≠ {0}, how many solutions will the system Ax=c have? Explain.
  11. M

    B Zero eigenvalue and null space

    Suppose ##T## is an operator in a finite dimensional complex vector space and it has a zero eigenvalue. If ##v## is the corresponding eigenvector, then $$ Tv=0v=0 $$ Does it mean then that ##\textrm{null }T## consists of all eigenvectors with the zero eigenvalue? What if ##T## does not have zero...
  12. Y

    Linear Algebra - Left Null Space

    Homework Statement I am given the follow graph and asked to find the left null space Homework EquationsThe Attempt at a Solution First I start by transpose A because I know that the left null space is the null space of the incidence matrix transposed. I then reduce it to reduce row echelon...
  13. Samuel Williams

    Linear Algebra - Transformation / operator

    Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...
  14. A

    Reason for just a 0 vector in a null space of a L.I matrix

    Hello Everyone, Can someone explain why do matrices with linearly independent columns have only 0 vector in their null space? Thanks
  15. A

    MHB Basis for column and null space

    Please help me with these three questions. I'm really struggling to understand these concepts and I think that with an understanding of these three, I will be able to tackle the rest before my test on Wednesday. Thank you. http://www.texpaste.com/n/g4rwmzzw 1) $$ A = \left[\begin{matrix} -6 &...
  16. Paul Shredder

    Structure of a Matrix With Empty Null Space

    Hi guys, I hope you are having a great day, this is Paul and, as you have seen in the title, that's what I'm looking for, let me explain: When you have a square matrix with empty null space, that is, the only solution to the equation Ax=0 (with dim(A)=n x n) is the vector x=0n x 1, means that...
  17. K

    Zero Dimensional Null Space (What's the meaning of this?)

    So a question on my linear algebra homework asks for the dimensions of Nul(A) and Col(A). Let A = \begin{pmatrix} -4 & -3\\ -1 &4\\ -3& -7 \end{pmatrix} I row reduced the above matrix to \begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix} Now, the T.A. for my section told us that to find the...
  18. Muthumanimaran

    Column space and null space

    Why it is important to know about Column space and Null spaces in Linear Algebra?
  19. M

    Linear Algebra, null space

    Homework Statement Construct a matrix whose null space consist of all linear combinations of: v1 = (Column matrix) <1 -1 3 2> v2 = (Column matrix) <2 0 -2 4>Homework Equations NS(A) = {x ε Rn I Ax =0} w = k1v1 + k2v2The Attempt at a Solution I'm...
  20. N

    Find a basis for the null space of the transpose operator

    Homework Statement Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##. Homework Equations Let ##T:V\to W## be a linear...
  21. K

    Finding the Null Space of a Matrix: A Guide to Solving for the Solution Set

    Homework Statement What is the null space of this matrices. |1 1 0 3 1| |0 1 -1 0 1| |1 1 3 0 1| The Attempt at a Solution I reduced it to rref using agumented matrice(each one equals to zero ) |1 0 0 4 0 0| |0 1 0 -1 1 0| |0 0 1 -1 0 0| and i get get x1=-x4 , x2= x4...
  22. I

    Guidance: Convex hull, null space and convex basis etc

    Hi friends! I am getting started with a research paper that discusses the closure properties of a robotic grasp. There are of lot of mathematical terms that confuse me like 'convex hull' , convex basis, convex combination of vectors, a free subset, nullspace etc. I might have studied some of...
  23. Daaavde

    Diagonalizable endomorphism has trivial null space

    Is it correct to state that a diagonalizable endomorphism has always kernel = {0}?
  24. D

    Finding a basis for the null space and range of a matrix

    Homework Statement ##S## is a linear transformation and ##\{u_{1},u_{2}\}## is a basis for the vector space. $$ S(u_{1})=u_{1}+u_{2}\\ S(u_{2})=-u_{1}-u_{2} $$ I would like to find a basis of the null space and range of ##S##.Homework Equations In my text, it says that the proper matrix...
  25. M

    Null Space of a Matrix and Its Iterates

    This might seem like a stupid question but would the null space of a matrix and its, say Gaussian elimination transforms, have the same null space. I guess, I am asking if this is valid: Let x be in N(A). Let A_{m} be some iteration of A through elimination matrices, i.e. A_{m} =...
  26. N

    Null space and eigenspace of diagonal matrix

    Homework Statement I am working on a problem where I made a matrix representation of a linear transformation and I am asked what is the eigenspace for a particular eigenvalue. Homework Equations The Attempt at a Solution The problem for me is, I came out with a diagonal...
  27. S

    MHB Row Space, Column Space and Null Space

    1.Construct a matrix whose null space consists of all linear combination of the vectors, v1={1;-1;3;2} and v2={2,0,-2,4} (v1,v2 are column vector).2.The equation x1+x2+x3=1 can be viewed as a linear system of one equation in three unknowns. Express its general solution as a particular solution...
  28. A

    A'A and A have the same null space

    I'm trying to prove that the null space of A'A is the null space of A, this is what I have so far, 1) A'Ax=0, non trivial solutions are a basis for the null space of A'A 2) x'A'Ax=0 3) (Ax)'Ax=0 4) Since (Ax)'A is a linear combination of the col's of A, we see that the null space of...
  29. L

    Dimension of the null space of A transpose

    So I'm given a matrix A that is already in RREF and I'm supposed to find the null space of its transpose. So I transpose it. Do I RREF the transpose of it? Because if I transpose a matrix that's already in RREF, it's no longer in RREF. But if I RREF the transpose, it gives me a matrix with 2...
  30. R

    Null space and 3x3 matrix

    Homework Statement Homework Equations The Attempt at a Solution I don't understand how they get the numbers on the right. This is a null space problem so the 3x3 matrix = 0. By my reckoning (1/3)x3 = 0 so x3 = 0. So then I try the second row. 2x3 = (3/2)x2 Divide both...
  31. E

    Give a matrix, B, so that it's null space is a given set of vectors

    Homework Statement Give a matrix B so that the subspace W defined in part b (W = (1,1,0,-2),(1,-1,1,6),(0,1,1,4)) can be written as W = N(B) where N(B) is the null space of B Homework Equations none that I know of, other than N(A) = {vectors x | Ax = 0} The Attempt at a Solution...
  32. J

    Mapping a matrix to the null space

    Homework Statement I am trying to run a model in matlab. D is a 2 by 3 matrix, Knowing that DL=0, which means L is mapped to the null space. Homework Equations How can i find L so that it is a 3 by 3 matrix with all its entries being one times a scalar. The Attempt at a Solution...
  33. G

    Finding Basis of Null Space and Range

    Homework Statement Prove T is a linear transformation and find bases for both N(T) and R(T). Homework Equations The Attempt at a Solution T:M2x3(F) \rightarrow M2x2(F) defined by: T(a11 a12 a13) (a21 a22 a23) (this is one matrix) = (2a11-a12 a13+2a12)...
  34. A

    Finding the basis of a null space

    Homework Statement The matrix is: -2 -2 -4 4 -1 1 2 -2 -1 0 -3 0 -4 1 -7 -2 I know the dimensions for the null space are 2 Homework Equations I know that to find the basis for a null space Ax=0, so I row reduced it and I got 1 0 3 0 0 1 5 -2 0 0 0 0 0 0 0 0 The Attempt...
  35. 3

    Finding the Null Space of a Matrix | Solving for x in Ax=0 | Linear Algebra

    Homework Statement Determine the null space of the following matrix: A = [1 1 -1 2 2 2 -3 1 -1 -1 0 -5] Homework Equations Ax=0 where x = (x_{1}, x_{2}, x_{3}, x_{4})^{T} The Attempt at a Solution If I put the system Ax=0 into augmented form: 1 1 -1 2 | 0 2 2 -3 1 | 0...
  36. S

    Range & Null space of A matrix

    Homework Statement Let x \in RN, y \in RM & A \in RMxN be a matrix. Denote the columns of A by Ak, k = 1,...,N. Let R(A) & N(A) be the range & null space of A respectively. a) How do the colmuns of A relate to the range of A? b) Your task is to find the solution to the problem y = Ax, where...
  37. P

    Nullity of Matrix A: Implications & Null Space Span

    For a matrix A, if its nullity is equal to 1, what is the implication of that? What spans its null space? Thanks a lot!
  38. A

    Finding a Linear Transformation T: R2 -> R2 with Equal Null Space and Range

    Homework Statement Give an example of a linear transformation T: R2 -> R2 such that the null space is equal to the range. Homework Equations null space and range The Attempt at a Solution I have been trying to come up with a solution but I cannot figure it out. What might be a...
  39. P

    Linear Algebra Null Space and Range

    give a basis for the range and the null space of T:P2(R) to P1(R) where for all p element of P2(R), T(p)=3p'' - p' I got the null space is {1} and the range is {x,x^2} but the answer says it should be {1,x} for the range. How can something be apart of the null space and the range if its...
  40. Y

    (n-1)-dimensional subspace is the null space of a linear functional

    Given that N is an (n-1)-dimensional subspace of an n-dimensional vector space V, show that N is the null space of a linear functional. My thoughts: suppose \alpha_i(1\leq i \leq n-1) is the basis of N, the linear functional in question has to satisfy f(\alpha_i)=0. Am I correct? Thanks
  41. T

    Easy calculation of basis of the null space

    Homework Statement find the basis of the nullspace of this matrix \begin{pmatrix} 1&1&1&-1 \\ 0&0&1&3 \end{pmatrix}Homework Equations The Attempt at a Solution i forget it. i first substitue 0 and 1 for last row but what about the first row? Substiute 0 and 1 again and this will give 4 basis...
  42. P

    Null Space and Eigenvalues/Eigenvectors

    Suppose I have a linear operator of dimension n, and suppose that this operator has a non-trivial null space. That is: A \cdot x = 0 Suppose the dimension of the null space is k < n, that is, I can find 0 < k linearly independent vectors, each of which yields the 0 vector when the linear...
  43. R

    Find a basis for the null space

    Homework Statement You're given two matrices (A and B). You want to find a basis for the space {x|x = Ay where By =0}. Homework Equations The Attempt at a Solution You're looking for all vectors x=Ay such that y is in the null space of B. So you're looking for a basis for only a part of...
  44. R

    Matrix ranks and null space

    Hi guys, I basically need help with matrices, I know all the basics about them like inverse, determinants, eigenvalues and eigenvectors and all but I need help in some topics like matrix rank, null space and all. I haven't read about them in any book so if you guys can post me links of...
  45. S

    Null space vs Col space dimension?

    I have a question in my linear algebra text that asks: Give integers p and q such that Nul A is a subspace of Rp and Col A is a subspace of Rq. What determines these values? Why are the values of p and q different between the Nul space and Col space? The matrix in question is a 3 x 4...
  46. C

    Understanding Null Space: A Layman's Explanation

    Can someone give me a layman’s terms explanation of what a null space is .
  47. M

    If n*n matrix, can row space ever be equal to null space?

    If n*n matrix, can row space ever be equal to null space? P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head.
  48. C

    Range and null space of T

    Given a linear transformation T from V to V, can we say that the range of T is in the space spanned by the column vectors of T. And we already know that the null space of T is the one spanned by the set of vectors that are orthogonal to the row vectors of T, then is there any general...
  49. S

    Help, null space projection

    Hello everyone, If I have a collection of data points (vectors), and x and y are two vectors among them. I want to project the data to a direction that the Euclidean distance between x and y is Maximally preserved. Then this direction should be the row space of (x-y)’, denoted as row( (x-y)’...
  50. V

    Linear Algebra - Column and Null Space - Take 2

    Homework Statement Ax=b where, A = 1 -1 ...-1 1 Homework Equations a) Find Null Space N(A) and Column Space C(A) b) For which vectors b does the system Kx=b have a solution? c) How many solution x does the system have for any given b The Attempt at a Solution a) For Null Space, I got x =...