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:
The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n1 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...
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}^3xy=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##...
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 GaussJordan Elimination
$\left(\begin{matrix}2&1&0\\1&0&0\\0&0&0\end{matrix}\right)$ $\implies$...
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 nonrow space component?"
Take a matrix
A = \begin{bmatrix}1 & 2\\ 3 &...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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%2C2%2C3%7D%2C%7B0%2C1%2C2%2C1%7D%7D
Homework Equations
Ax = b (where x and b are vectors and A...
The problem is attached, I did parts 13, 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...
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...
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...
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?
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...
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...
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]
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...
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...
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...
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...
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]...
Homework Statement
It is number three on the following page.
http://people.math.carleton.ca/~mezo/A3math110211.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 +...
Homework Statement
Find the kernel of the matrix:
http://img256.imageshack.us/img256/9015/53369959.jpg
The Attempt at a Solution
So I rowreduce it and get:
[PLAIN][PLAIN]http://img812.imageshack.us/img812/1391/97980793.jpg
The system of equations the rowreduced form equals 0.
So I set...
Homework Statement
Suppose a 3 x 5 matrix A has rowreduced 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...
Homework Statement
How to verify that the nullspace is orthogonal to the row space of B?
I have inserted the screenshot 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...
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?
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...
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...
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...
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)...
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...
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?
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?
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...
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]
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...
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...
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...
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)...