What is Linear transformations: Definition and 200 Discussions
In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping
V
→
W
{\displaystyle V\to W}
between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.
If a linear map is a bijection then it is called a linear isomorphism. In the case where
V
=
W
{\displaystyle V=W}
, a linear map is called a (linear) endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that
V
{\displaystyle V}
and
W
{\displaystyle W}
are real vector spaces (not necessarily with
V
=
W
{\displaystyle V=W}
), or it can be used to emphasize that
V
{\displaystyle V}
is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.
A linear map from V to W always maps the origin of V to the origin of W. Moreover, it maps linear subspaces in V onto linear subspaces in W (possibly of a lower dimension); for example, it maps a plane through the origin in V to either a plane through the origin in W, a line through the origin in W, or just the origin in W. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.
In the language of category theory, linear maps are the morphisms of vector spaces.
For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix.
For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself.
The expressions have similar form except for the order of the inverses. Is there there any connection...
In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix
to correspond to the d/dx linear transformation...
Let C2x2 be the complex vector space of 2x2 matrices with complex entries. Let and let T be the linear operator onC2x2 defined by T(A) = BA. What is the rank of T? Can you describe T2?
____________________________________________________________
An ordered basis for C2x2 is:
I don't...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis.
When we switch...
ok we are supposed to go to here
Find 3 different matrices that reflect the following transformations, report the matrix, the determinant, and the eigenvalues.
1. Rotation by $\dfrac{\pi}{4}$
2. Shear along $x$ by a factor of $k$
3. Reflection by the line $\theta$
there are some more but the...
Let ##V## be a real vectorspace of finite dimension ##n##. Let ##L, K:V \rightarrow \Re## be linear transformations so that ##ker(L) \subset ker(K)##. Then there's a parameter ##\lambda \in \Re## so that ##K=\lambda L##
a) Show that ##K=\lambda L## holds when ##K=0##.
b) Suppose that ##K...
At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following:
" ... ... Each matrix A in this group determines an invertible linear transformation f_A: \mathbb{R} \to \mathbb{R} defined by f_A(x) = x A^t ... ... "I know that one may define...
Summary:: linear transformations
Hello everyone, firstly sorry about my English, I'm from Brazil.
Secondly I want to ask you some help in solving a question about linear transformations.
Here is the question:Consider the linear transformation described by the matrix \mathsf{A} \in \Re...
Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A).
Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?
(I can do this with numbers but the trig is throwing me).
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am reading Chapter 8: Differentiable Maps ... ... and am currently focused on Section 8.1 Linear Algebra ... ...
I need some help in order to fully understand some remarks by Browder in Section 8.1, page...
Given w = T (v), where T is a linear transformation and w and v are vectors, why is it that we can write any coefficient of w, such as w1 as a linear combination of the coefficients of v? i.e. w1 = av1 + bv2 + cv3
Supposably this is a consequence of the definition of linear transformations, but...
Good afternoon people.
So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if:
T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to...
ok this is a clip from my overleaf homework reviewing
just seeing if I am going in the right direction with this
their was an example to follow but it also was a very different problem
much mahalo
I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ...
I need some help in order to fully understand Example 6.76 ... ...
Example 6.76 reads as follows:
My question or...
Hello,
I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the...
Homework Statement
Homework Statement
(a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let
##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}##
Prove that ##S## and ##A## are subspaces and...
I have learned that for matrix theory, for ##A \vec{x} = \vec{b}##, if there exists a particular solution ##p##, then every solution looks like ##p+k##, where ##k \in \ker A##.
Can someone help me find material on this online, but only in the context of general linear transformations? For...
Homework Statement
How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##?
Homework Equations
Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
The "Operator Norm" for Linear Transformations ... Browder, Lemma 8.4, Section 8.1, Ch. 8 ... ...
I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...
Homework Statement
given that X is an n × p matrix with linearly independent columns.
And $$X^∗ = XA$$ where A is an invertible p × p matrix.
a)
Show that: $$X^*{({X^*}^TX^*)^-}^1{X^*}^T = X{(X^TX)^-}^1X^T$$
b)
Consider two alternative models
$$M : Y = Xβ + ε$$ and $$M^∗ : Y = X^∗β ^∗ +...
Homework Statement
"Show that every subspace of ##ℝ^n## is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace ##W## consists of only the zero vector or is all of ##ℝ^n##, ##W## is the set of solutions to ##IX=0## or ##0_vX=0##, respectively.
Assume ##W## is...
Homework Statement
"Determine whether the function ##T:M_{2×2}(ℝ)→ℝ## defined by ##T(A)=det(A)## is a linear transformation.
Homework Equations
##det(A)=\sum_{i=1}^n a_{ij}C_{ij}##
The Attempt at a Solution
I'm assuming that it isn't a linear transformation because ##det(A+B)≠det(A)+det(B)##...
Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after show my...
Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after write my...
We learned that the condition of a linear transformation is
1. T(v+w) = T(v)+T(w)
2. T(kv) = kT(v)
I am wondering if there is any transformation which only fulfil either one and fails another condition. As obviously, 1 implies 2 for rational number k.
Could anyone give an example of each...
Homework Statement
We're given some linear transformations, and asked what the null space, column space and row space of the matrix representations tell us
Homework EquationsThe Attempt at a Solution
I know what information the column space and null space contain, but what does the row space of...
Homework Statement
Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$
a) Show ##T## is a linear transformation from ##C## to ##C##.
b)What...
My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
I am revising the basics of linear transformations and trying to get a thorough understanding of linear transformations and their matrices ... ...
At present I am working through examples and exercises in Seymour Lipshutz' book: Linear Algebra, Fourth Edition (Schaum Series) ... ...
At...
Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ...
Now ...
Suppose we have a linear transformation T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2 , say ...
Suppose also that \mathbb{R}^3 has basis B and \mathbb{R}^2 has basis B'...
I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ...
I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.6 Constructing Linear Transformations ...
I need help with a basic...
I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ...
I am focused on Section 2.1 Introduction to Linear Transformations ... ...
I need help with understanding Theorem 2.7 ...
Theorem 2.7, its proof and some remarks read as follows:I am having considerable trouble...
Homework Statement
Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where
B=
\begin{bmatrix}
3 & -2 \\
2 & -2 \\
\end{bmatrix}
Find rank,defect and one basis of an image and kernel of linear operator...
Homework Statement
[/B]
Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2Homework Equations
[/B]
See the above notesThe Attempt at a Solution
[/B]
I am making a pig's ear out of this one.
I think I can get e1 to –e2
3 -1
1 -3
but as far as getting it to reconcile a matrix like...
Assume the mapping T: P2 -> P2 defined by:
T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2
is linear.Find the matrix representation of T relative to the basis B = {1,t,t2}
My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not...
Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer.
I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
Hi can anyone give me some hints with this question thanks
A = \begin{pmatrix} 3 & -2 &1 & 0 \\ 1 & 6 & 2 & 1 \\ -3 & 0 & 7 & 1 \end{pmatrix}
be a matrix for T:ℝ4→ℝ3 relative to the basis
B = {v1, v2, v3, v4} and B'= {w1, w2, w3}
v1 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \end{pmatrix}
v2 =...
Homework Statement
Let V be a vector space over a field F and let L and M be two linear transformations from V to V.
Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V .The Attempt at a Solution
I presume it's a simple question, but it's one of those where you just don't...
Homework Statement
Give a thorough explanation as to why a linear transformation:
with a standard matrix A CANNOT be one to one.
Homework Equations
The Attempt at a Solution
I think I have figured this one out, but I was hoping somebody could confirm whether this example is sufficient...
Homework Statement
Consider a 2x2 matrix A with A2=A.
If vector w is in the image of A, what is the relationship between w and Aw?
Homework Equations
Linear transformation T(x)=Ax
Image of a matrix is the span of its column vectors
The Attempt at a Solution
I know that vector w is one of the...
This is the new topic we are going over in my Linear Algebra class and I am completely lost in how it works. From what I gather from the book, the lecture, and other sources amongst the internet, there are supposed to be a rule set shard with the transformation. The problem being is I cannot...
I have a small confusion about functions and variables. So, on doing a bit of reading, a linear transformation is a function that maps inputs from one vector space to another.
So, let us take for example a simple rotation matrix. This matrix takes a point in 2D space and maps it to another...
Homework Statement
Hey PF, I'm here again asking about linear transformations, ha ha.
Let C={(x,y) \in \mathbb R2 | x²+y²≤1} a circle of radius 1 and consider the linear transform
T:\mathbb R2→\mathbb R2
(x,y) \mapsto (\frac{5x+3y}{4},\frac{3x+5y}{4})
Find all values of a natural n for...