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:
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...
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...
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...
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.
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...
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...
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...
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.
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...
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...
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...
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) $$ A = \left[\begin{matrix}
-6 &...
Hi guys, I hope you are having a great day, this is Paul and, as you have seen in the title, that's what I'm looking for, let me explain:
When you have a square matrix with empty null space, that is, the only solution to the equation Ax=0 (with dim(A)=n x n) is the vector x=0n x 1, means that...
So a question on my linear algebra homework asks for the dimensions of Nul(A) and Col(A).
Let A =
\begin{pmatrix}
-4 & -3\\
-1 &4\\
-3& -7
\end{pmatrix}
I row reduced the above matrix to
\begin{pmatrix}
1 & 0\\
0 & 1\\
\end{pmatrix}
Now, the T.A. for my section told us that to find the...
Homework Statement
Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##.
Homework Equations
Let ##T:V\to W## be a linear...
Homework Statement
What is the null space of this matrices.
|1 1 0 3 1|
|0 1 -1 0 1|
|1 1 3 0 1|
The Attempt at a Solution
I reduced it to rref using agumented matrice(each one equals to zero )
|1 0 0 4 0 0|
|0 1 0 -1 1 0|
|0 0 1 -1 0 0|
and i get get x1=-x4 , x2= x4...
Hi friends!
I am getting started with a research paper that discusses the closure properties of a robotic grasp. There are of lot of mathematical terms that confuse me like 'convex hull' , convex basis, convex combination of vectors, a free subset, nullspace etc. I might have studied some of...
Homework Statement
##S## is a linear transformation and ##\{u_{1},u_{2}\}## is a basis for the vector space.
$$
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...
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} =...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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.
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...
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)’...
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 =...