# 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:

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

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1. ### 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...

25. ### 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. ### 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. ### 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 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### (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. ### 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. ### 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. ### 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. ### 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. ### 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. ### Understanding Null Space: A Layman's Explanation

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