# What is Nullspace: Definition and 59 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. ### Intro to Linear Algebra - Nullspace of Rank 1 Matrix

The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
2. ### Check of a problem about nullspace

Homework Statement Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##. Find a matrix ##A## associated to a linear map ##f:\mathbb{R}^3\rightarrow\mathbb{R}^3## through the standard basis such that its nullspace is ##V##...
3. ### MHB Interpreting & Solving Nullspace

What is the interpretation of this nullspace? How to write the solution in parametric form if possible? $N( \left(\begin{matrix}2&-1&0\\1&0&0\\0&0&0\end{matrix}\right))$ Using Gauss-Jordan Elimination $\left(\begin{matrix}2&-1&0\\1&0&0\\0&0&0\end{matrix}\right)$ $\implies$...
4. ### Row and null complements of x; need clarity....

I've managed to distill the rambling into just this question, posted here and at the end of my digressive thoughts as well: "Will we always be able to split x up in such a way that we have a nullspace component and a non-row space component?" Take a matrix A = \begin{bmatrix}1 & 2\\ 3 &...
5. ### Very basic Q about solns to y" = y

Wolfram and the Linear Algebra text I'm currently working on, give the two possible solutions of \frac{d^2y}{dx^2}=y as being e^{x} and e^{-x}, or rather, constant multiples of them. Here wolfram agrees: http://www.wolframalpha.com/input/?i=d^2y/dx^2=y My question is, why isn't y = e^{x} + x...
6. ### Column space and nullspace relationship?

I have just been studying Nullspaces... I want to make the following summary, will it be correct? C(A) is all possible linear combinations of the pivot columns of A. N(A) is all possible linear combinations of the free columns of A (if any exist). edit: I have a feeling these are...
7. ### Nullspaces relation between components and overall matrix

Homework Statement If matrix ## C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right]## then how is N(C), the nullspace of C, related to N(A) and N(B)? Homework Equations Ax = 0; x = N(A) The Attempt at a Solution First, I thought that the relation between A and B with C is ## C = A...
8. ### How they found the left nullspace in each of these examples

Homework Statement Part b) http://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture19_with_Examples.pdf For B Left nullspace is solution to A ^ T times Y =0 So we have a free variable for the third row so don't we have infinitely many solutions as x3 could be anything? In...
9. ### Is N(A) a Subset of N(A^t A)?: Proving Inclusion for Matrix Nullspaces

Homework Statement Given matrix A (size m x n), prove N(A) is subset of N( A^t A). A^t is matrix A transposed. Homework Equations The Attempt at a Solution My assumption is m < n, using definition of nullspace, I ended up with N( A^t A) = a set of zero vector, while N(A) is...
10. ### Nullspace of A transpose x: A Geometric Interpretation

What does ATx=0 means? Does this notation means if A = [3,2;1,2;4,4], then, AT = [3,1,4;2,2,4] and ATx [x1;x2;x3] = 0? The nullspace of the transposed of the matrix A?
11. ### Nullspace Matrix Homework: Struggling to Find an Example Vector

Homework Statement Let A be the matrix: [3,3,-2,0;-3,-3,3,-2] a) An example of a vector in the nullspace of A is b) An example of a vector NOT in the nullspace of A is Sorry guy but I'm really STRUGGLING The Attempt at a Solution a) I found x1 ,x2,x3,x4 = -x2+4/3x4, x2...
12. ### Find a matrix with S as its nullspace

Homework Statement Let S be the subspace of R4 given by the solution set of the equations -b + c + d = a - 3 c and -a - 2 d = d = a - c Find an example of a matrix for which S is the nullspace. Homework Equations Ax=0 The Attempt at a Solution I have found that the...
13. ### Understanding the Nullspace of a Matrix for Subspace U in R4

Homework Statement Let U be the subspace of R4 given by: U = the nullspace of the matrix [0 0 2 4 0 3 -4 2] The Attempt at a Solution let v = (v1,...v4) and w = (w1...w4) (0,0,2,4) = (λ1v1 + λ2v2 + λ3v3 + λ4v4) (0,3,-4,2) = (λ1w1 + λ2w2 + λ3w3 + λ4w4) I haven't come...
14. ### Direct sum of nullspace and range

Is this true? I am studying direct sums and was wondering if the following statement holds? It seems to be true if one considers the proof of the dimension theorem, but I need to be sure, so I can steer my proof toward a particular direction. ## N(T) \bigoplus R(T) = V ## where ##V## is the...
15. ### Nullspace of a square matrix A and A^2 are related?

Homework Statement Say that A is a square matrix. Show that the following statements are true, or give a counter example: a) If x is in the nullspace of A, then x is in the nullspace of A2 b) If x is in the nullspace of A2, the x is in the nullspace of A. Homework Equations The...
16. ### Construct a matrix whose nullspace consists of all combinations [ ]

Homework Statement Construct a matrix whose nullspace consists of all combinations of (2,2,1,0) and (3,1,0,1). Apparently, the answer is: http://www.wolframalpha.com/input/?i=%7B%7B1%2C0%2C-2%2C-3%7D%2C%7B0%2C1%2C-2%2C-1%7D%7D Homework Equations Ax = b (where x and b are vectors and A...
17. ### Why is that Nullspace of A is subset of nullspace of A^T*A

Why is that Nullspace of A is subset of nullspace of A^T*A let's say that A is m*n matrix
18. ### Nullspace of matrix A and the nullspace of A^T*A

Homework Statement A is matrix m*n show that nullspace of A is the subset of nullspace of A^T*A Homework Equations The Attempt at a Solution
19. ### Basis, Nullspace, Linear transformtion

The problem is attached, I did parts 1-3, but I am having trouble with part 4. This is what i was planning on doing for part 4 (my teacher said this wasn't the correct method): set T(v)=0 and solve the augmented matrix 1 0 -1 1 0 2 1 -2 4 0 3 1 -1 7 0 rref gives 1 0 0 2 0 0 1 0 2 0 0 0 1 1 0...
20. ### Finding a Basis for the Nullspace of a 2x2 Matrix Transformation

The problem is attached. I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1Applying the transformation to each of these gives 0 0 0 0 0 2 0 0 0 0 -2 0 0 0 0 0 respectively. Now this is where I get stuck. How do I find a...
21. ### Finding basis for nullspace of transformation

T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...
22. ### Finding the Nullspace of an Invertible 3x3 Matrix

Let's say you have a 3x3 matrix and it's invertible. Let's call it A If you were to find a basis for the nullspace of A, would the basis just be the original 3 column vectors of A?
23. ### Equivalence of the nullspace and eigenvectors corresponding to zero eigenvalue

Suppose a square matrix A is given. Is it true that the null space of A corresponds to eigenvectors of A being associated with its zero eigenvalue? I'm a bit confused with the terms 'algebraic and geometric multiplicity' of eigenvalues related to the previous statement? How does this affect the...
24. ### Is matrix A in the form of RREF?Is Matrix A in RREF form?

I attached 2 problems. For problem #1. I want to make sure I'm on the right track, to find the span of Null(A), i need to put matrix A in RREF form. By doing so I get x1=-2t x2=-t x3=s x4=u (using u because I'm using t to denote transpose) where x1 to x4 is for each respective column...
25. ### Understanding the Nullspace of Eigenspaces

Homework Statement Homework Equations The Attempt at a Solution I don't understand how u_1 = [1 -1]^T ? By my reckoning u_1 = \frac{v_1}{\parallel v_1 \parallel} which is \frac{-2}{\parallel -2+1 \parallel}, \frac{-2}{\parallel -2+1 \parallel} which is [-2, -2] not [1, -1]
26. ### Finding rank(range) and nullspace of a matrix

Homework Statement Trying to figure out the rank and nullspace of the matrix of matrix A and B: A= 1 0 5 4 1 4 B= 1 0 1 5 4 9 2 4 6 Homework Equations I used the Guass elimination on both The Attempt...
27. ### Subspace, vectorspace, nullspace, columnspace

I am wondering how to organise all of those concepts in my head. should i think of it like: subspace > vectorspace > nullspace, columnspace kind of like columnspaces and nullspaces are valid vectorspaces, and all of those are valid subspaces. is a vector space a columnspace? except its...
28. ### Find the left nullspace of matrix

Homework Statement B= \displaystyle\left[ {\begin{array}{*{20}{c}} 1&0&-2&1 \\ 1&2&-2&3 \\ -2&1&3&0 \end{array}} \right] Find: 1. The nullspace of B. 2. The left nullspace of B. The attempt at a solution I was able to find the nullspace of B. but i can't figure out why the left...
29. ### The nullspace of a transposed matrix

Let says I have a matrix A with m rows and n columns, with m<n, from which I compute the null space. If the rank of A is smaller than m, then the null space of the transpose of A also exists. Is there any relation between the null space of a matrix and the null space of the transposed matrix...
30. ### Splitting a vector into a rowspace component and a nullspace component

Homework Statement Find a basis for the orthogonal complement of the row space of A: A = [1 0 2 1 1 4] Split x = (3,3,3) into a row space component xr and a nullspace component xn. The Attempt at a Solution For the first part of the problem I took A to RREF R = [1 0 2 0 1 2]...
31. ### Proof that the nullspace is closed in addition/multiplication

Homework Statement It is number three on the following page. http://people.math.carleton.ca/~mezo/A3math1102-11.pdfHomework Equations No idea. The Attempt at a Solution I have no idea how to incorporate the kj. Best I could reason through this is supposing: b1 ∈ N(A) , c1 ∈ N(A) Ab1 +...
32. ### Understanding nullspace (kernel) of a matrix

Homework Statement Find the kernel of the matrix: http://img256.imageshack.us/img256/9015/53369959.jpg The Attempt at a Solution So I row-reduce it and get: [PLAIN][PLAIN]http://img812.imageshack.us/img812/1391/97980793.jpg The system of equations the row-reduced form equals 0. So I set...
33. ### Nullspace, Column Space, and solution of system given only rref(A)

Homework Statement Suppose a 3 x 5 matrix A has row-reduced echelon form: [[1 2 0 0 5] [0 0 1 0 4] [0 0 0 1 3]] a. Describe NS(A) b. Describe CS(A) c. Suppose . [[2] . [3] [[-2] A [5] = [4] = b . [1] [3]] . [9]] To be clear, that's the original matrix A times the...
34. ### How to verify that the nullspace is orthogonal to the row space?

Homework Statement How to verify that the nullspace is orthogonal to the row space of B? I have inserted the screen-shot of the problem below: http://i29.fastpic.ru/big/2011/0918/10/ca341692cc37b831143f5fe32351db10.jpg Homework Equations Nullspace and orthogonality.The Attempt at a Solution I...
35. ### Simple Q about calculating Nullspace

When calculating the nullspace of a n x n matrix, after i have reduced the matrix to row echelon form, DO ALL MY PIVOTS HAVE TO BE 1 BEFORE i can distinguish the free variables, and then calculate the vectors that satisfy Ax=0?
36. ### Vector spaces homework question (rowspace and nullspace)

Homework Statement Write x=(6,-1,-2)T as x=y+z where y belongs to null A and z belongs to row A A=[1,3,1;2,6,2;-2,-5,0;1,4,3] Homework Equations The main question asks to find all the fundamental subspaces and their dimensions, which I have already found, and then asks me to find the...
37. ### How Do You Determine the Basis, Dimension, and Rank of Vector Sets?

Homework Statement for the set of vectors: v_1 = 1, -2, 0, 0, 3 v_2 = 2, -5, -3, -2, 6 v_3 = 0, 5, 15, 10, 0 v_4 = 2, 6, 18, 8, 6 (a) find a basis for the set of vectors and state the dimension of the space spanned by these vectors, what is the rank of this matrix? (b) construct a matrix whose...
38. ### Proof That A & A^T Have the Same Nullspace (Kernel)

Hello, can you help me with the proof? If A is normal A^TA=AA^T then A and A^T have the same nullspace (kernel). And ||Ax||=||A^Tx|| Thank you.
39. ### Image and nullspace bases of a linear transformation

Homework Statement Let T be the linear transformation T: M2x2-->M2x2 given by T([a,b;c,d]) = [a,b;c,d][0,0;1,1] = [b,b;d,d] Find bases (consisting of 2x2 matrices) for the image of T and the nullspace of T. Homework Equations Standard basis of a 2x2 matrix...
40. ### Differnece between the Kernel and the Nullspace?

What is the distinct difference between the kernel and the null space? They both have the same definition, namely, Ax=0.
41. ### What is the nullspace of a 3x3 complex matrix?

Homework Statement I have the 3x3 matrix C=(1,-1,1; 2,0,1+i; 0,1+i,-1) and I want to find its nullspace (a set of vectors that span that subspace). The Attempt at a Solution So first I have reduced the matrix to row echelon form and I got this matrix: (1,-1,1; 0,1,-0.5+0.5i; 0,0,0)...
42. ### Linear transformation given a nullspace and a solution space.

Homework Statement Find if possible a linear transformation R^4-->R^3 so that the nullspace is [(1,2,3,4),(0,1,2,3)] and the range the solutions to x_1+x_2+x_3=0. Homework Equations - The Attempt at a Solution So I thought I should start with trying to find what kind of matrix we...
43. ### Finding x* in the Nullspace of A: What is its Significance?

Does anyone know how to approach this problem? Let A be an m×n matrix of rank m, where m<n. Pick a point x in R^n, and let x∗ be the point in the nullspace of A closest to x. Write a formula for x∗ in terms of x and A. What exactly is the significance of the point x* in the nullspace of A?
44. ### What is the optimal point in the nullspace of A using Lagrange multipliers?

Does anyone know how to approach this problem? Let A be an m×n matrix of rank m, where m<n. Pick a point x in R^n, and let x∗ be the point in the nullspace of A closest to x. Write a formula for x∗ in terms of x and A. What exactly is the significance of the point x* in the nullspace of A?
45. ### Geometric description of the nullspace

Homework Statement general form of solutions to Ax=b Consider matrix A= {[ 2 -10 6 ] [ 4 -20 12 ] [ 1 -5 3 ]} Find a basis for the nullspace of A. Give a geometric description of the nullspace of A. The Attempt at a Solution I found the...
46. ### Basis for the nullspace of this matrix

this is apparently "really simple", but I just don't know how to do it from the examples I have and I feel like a moron... what's the basis for the nullspace of this matrix [ 2 3 1] [ 5 2 1] [ 1 7 2] [ 6 -2 0]
47. ### Basis of Nullspace: Linear Algebra & Differential Equations

I am in a linear algebra and differential equations course and have recently been learning how to find a basis for a nullspace, row space, or column space. However, I am EXTREMELY confused by a solution to a question in my textbook. The question asks to find the basis for the null space of a...
48. ### Having problems understanding nullspace

Homework Statement If V is the subspace spanned by (1,1,1) and (2,1,0), find a matrix A that has V as its row space. Find a matrix B that has V as its nullspaceHomework Equations Ax = 0 for a nullspaceThe Attempt at a Solution So straight off the bat, I think I can solve the first part. Should...
49. ### [Linear Algebra] Nullspace equals Column space

Homework Statement Why does no 3 by 3 matrix have a nullspace that equals its column space? Homework Equations NA The Attempt at a Solution A = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \] C(A) = \begin{bmatrix} 1 \\ 0 \\ 0...
50. ### Nullspace of Matrix H: Proving Basis with Independent Rows

Note: I don't know LaTeX that well, hence I have done my working in the images. Homework Statement Show that the rows of G are a basis for the null space of H (part of this question will be to show the independence explicitly)...