Null space Definition and 72 Threads

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:




ker

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

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  1. V

    Linear Algebra - Column and Null Space

    Homework Statement Ax=b where, A = 2 -1 ...-1 2 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...
  2. G

    Proving Disjoint Range & Null Space of Linear Operator T

    Homework Statement Given a linear operator T, show that if rank(T^2)=rank(T), then the range and null space are disjoint. So I know that I can form a the same basis for range(T^2) and range(T), but I'm not sure where to go from there.
  3. P

    Linear Algebra: Null Space and Dimension

    Homework Statement Prove that dim(nullA) = dim(null(AV)) (A is a m x n matrix, V is a n x n matrix and is invertible Homework Equations AX=0 and AVX = 0 Null(AV) = span{X1,..Xd} Null(A) = span{V-1X1,.., V-1Xd} The Attempt at a Solution so you need to prove that...
  4. J

    What is the general algorithm for computing the null space of a 3x3 matrix?

    There are 2 issues I want to talk about in this post. (1) General algorithm for gauss-jordan elimination computation of null space (2) Closed form solution to 3x3 null space Following the example here, https://en.wikipedia.org/wiki/Kernel_(linear_algebra) I thought a general algorithm to...
  5. R

    Linear Transformation and Null Space

    Homework Statement Let (u,v,w) be a basis for vector space V, and let L be a linear transformation from V to vector space W. If (L(u),L(v),L(w)) is linearly dependent, then dim(Null Space(L)) > 1. Homework Equations The Attempt at a Solution I don't see why dim(Null...
  6. A

    What is the null space basis and dimension of A in R^5?

    Homework Statement find a basis of the null space N(A) in R^5 of the matrix A = 1 -2 2 3 -1 -3 6 -1 1 -7 2 -4 5 8 -4 in M3*5 (R) and hence determine the dimension Homework Equations The Attempt at a Solution i found that A= 1 -2 2 3 1 0 0 1/5 2/5 -2/5 0 0 0 0 0 by...
  7. Q

    What Does It Mean When Col A Is a Subspace of the Null Space of A?

    I am just wondering what is meant when someone says the Col A is a subspace of null Space of A. What can you infer from that? Also what is a null space of A(transpose)A How do they relate to A? Are there theorems about this that I can look up?
  8. F

    Proving Vector Space of U is Null Space of T

    got a question show that the null space of T is a vector space of U given the mapping T:U->V i know that null space or kernal of T is kerT={uEU: T(u)=0} and is a subset of U but don't have a clue where to start applying this to my question?
  9. M

    Basis for null space, row space, dimension

    Homework Statement What are the basis for the row space and null space for the following matrix? Find the dimension of RS, dim of NS. [1 -2 4 1] [3 1 -3 -1] [5 -3 5 1] Homework Equations dim RS + dim NS = # of columns The Attempt at a Solution I reduced the matrix into...
  10. M

    Finding null space of a given matrix

    Homework Statement Find null space of A, NS(A) and sketch NS(A) in R2 or R3. A = [1 3 2; 2 6 4] Homework Equations AX = 0 The Attempt at a Solution I know the second row is twice the first one. I tried to solve for x1, x2 and x3 putting everything in the form of AX =0. I did...
  11. A

    Showing null space and range are invariant

    If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S. I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?
  12. K

    Finding a Basis of the Null Space of a Matrix A in R^5 | SOLVED

    [SOLVED] basis of a null space Homework Statement Find a basis of the null space N(A)\subsetR^5 of the matrix A= 1 -2 2 3 -1 -3 6 -1 1 -7 2 -4 5 8 -4 \inM3x5(R) and hence determine its dimension Homework Equations The Attempt at a Solution...
  13. S

    What is the Dimension of the Null Space for Matrix A?

    Find the Dimension of the null space of the given matrix A: | 1 3| |-2 -6| The Attempt at a Solution I honestly don't know how to work this at all. I think I'm confused as to what Null Space actually is, so that's making this a difficult problem to understand. please help.
  14. I

    Finding Null Space of a Matrix with Trigonometric Equations

    I need to find the null space of: \dotx \left(\begin{array}{cc}cos(\beta)-1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)-1\end{array}\right) so: \dotx \left(\begin{array}{cc}cos(\beta)-1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)-1\end{array}\right) \binom{x}{y} = 0...
  15. D

    Projection into the left null space

    Homework Statement I am trying to find the matrix M that projects a vector b into the left nullspace of A, aka the nullspace of A transpose. Homework Equations A = matrix A ^ T = A transpose A ^ -1 = inverse of A e = b - A x (hat) e = b-p I know that the matrix P that projects...
  16. X

    Null space of A = null space of A'A?

    Hihi I've been working on this problem for some time: if A is a (m x n) matrix, and A' denotes its transpose, then the null space of A is equal to the null space of A'A. Is this always true? I thought of a proof for the special case where A is given in reduced row echelon form, but fail to...
  17. A

    Understanding Null Space: A Plain English Guide

    could someone kinda explain in plain english what null space is? @@
  18. T

    What is the null space of TΦ for Φ = x over the interval [0,1]?

    Let V denote the vectore space of continuously differentiable functions, ƒ, over the interval [0,1] such that ƒ(0)=0. Suppose Φ is-contained C∞ [0,1] (set of infinitely differentiable functions over the interval [0,1]) and define the operator TΦ:V→R:ƒ→∫ƒ'(x)Φ(x)dx 0,1 Describe the null space...
  19. M

    Finding the null space, matrix fun wee

    Hello everyone I'm confused on finding the null space on this problem: the matrix is: 0 2 0 -5 0 1 4 0 0 0 1 0 0 0 0 1 null(A) = 2b - 5d = 0 b + 4c = 0; c = 0 d = 0; b = 0; a = ? You don't know what a is, so I'm quite confused. Any help?
  20. K

    Solving Null Space Confusion in R^3

    The question is: For each of the following subspaces, find the dimension and a basis: {(x,y,z) are elements of R^3: 7x - 3y + z = 0} I had actually posted about this before, but I'm confused as to what the Null space is here. So, z = -7x + 3y, so there is one dependent variable and two...
  21. K

    Linear Algebra: Basis for the Null Space of a Matrix

    I think I understand how to do this, but I wanted to double check my work. I have to find the basis of the null space for the matrix: 1 0 2 0 0 7 So I knew that the basis of the image had two dimensions and a null space of one. The first and third columns are linearly independent (or at...
  22. T

    Question about the Null Space for this Zero Matrix

    How can I determine the null space for the 2 x 6 zero matrix as precisely as I can? Clearly N(A) = {x: Ax = 0, x in R^n}, So if A is this 2x6 matrix, wouldn't virtually any vector x that is in R^6 work? This is supposed to be a "conceptual" problem, and I KNOW it can't be this easy for...
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