# Questions tagged [linear-transformations]

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: http://en.m.wikipedia.org/wiki/Linear_map)

6,398 questions
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### Diagonalization of a matrix by a rotation matrix

Given is the following matrix $$A := \begin{pmatrix} 3x & 4x & 0 \\ 4x & 3x & 0 \\ 0 & 0 & 5x \end{pmatrix}$$ whereas $x \in \mathbb{R}$ is a constant. This matrix can be ...
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### [Linear Algebra][Recursive Sequence] A general function form from recursive sequence

I'm searching method to create a general form of function from recursive sequence. I've tried to use the linear fractional transformation, however I don't know whether this is a good idea to ...
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### Obtaining the matrix representation of a linear transformation [closed]

My question is about the answer here. Here, in this answer why: $$Av_j = Ae_j.$$
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### Numerical analysis of the terms of the Volterra series of high orders [closed]

The Volterra series contains a number of terms of order 2 and above. Each of the addends is a generalization of the convolution integral of 2 and higher output signals with 2 and higher output signals,...
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### Composition of transformations of a plane

How do I find a composition of linear transformations $$T_a\circ R_A^α\circ T_b$$?
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### Viewing the columns of a matrix as vectors, why do row operations not change the determinant?

Viewing the determinant as the amount of space enclosed by the columns of a matrix, it's geometrically clear to see why column operations don't change the determinant. All adding one column to ...
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### Linear algebra- matrix of rank $r$ as product of two matrices of rank $r$

Let $A$ be $m\times n$ matrix of rank $r$ then show that $A$ can be written as product of matrices of order $m\times r$ and $r\times n$. I know how to prove the above relation if $r=1$ but don't know ...
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### How exactly does Hahn-Banach theorem explain duality of vector spaces?

Serge Lang's Linear Algebra textbook just introduced me to the concept of dual space in very formal terms: space of all functional transformations having co-domain as $1$-dimensional vector space over ...
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### Does intersection of invariant subspace distribute over direct sum of invariant subspace found by minimal polynomial?

Let $T: V\to V$, and $V$ is of finite dimension. $M$ is the minimal polynomial $M = M_1 \cdot M_2 \cdot\dots\cdot M_k$. $M_i$ monic polynomial. $V = W_1 \oplus W_2 \oplus \dots \oplus W_k$. It's ...
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### Existence of Annihilating Polynomial for a Linear Operator over a fine dimensional vector space.

How do I show the existence of Annihilating Polynomial for a Linear Operator $T:V\to V$ over a finite dimensional vector space $V$. I had an idea that $\mathbb{L}(V,V)$ is a vector space of dimension ...
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### Why is orthogonal projection not always multiplication by a diagonal matrix?

Suppose I have a vector $v$ which I want to orthogonally project onto a subspace $S.$ The subspace is defined as the space spanned by the columns of a matrix $A.$ I could do this by constructing a ...
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### Insight on the polar decomposition of a shear?

I recently learned it, and really love the polar decomposition of a matrix, because it was the first time I actually could picture what it meant to "apply a transformation to space" (a phrase I kept ...
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### Number of Linear Transformations having same Kernel space dimensions

Let T:V->W. Is it possible to have multiple T's that have the same dim(ker(T)) dimensions of the kernel of T ? If not why ?
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### Number of zeros in $A^k$

Given a nilpotent matrix $N$ of order $n\times n$ such that $N^k=0$. Then i know that $m_{N}(x)=x^k$ and $c_{N}(x)=x^n$. This means that $N^k$ is a zero matrix. My question now is can we deduce how ...
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I'm studying adjoints, and I'm confused as to how I prove this. I have a definition of a self-adjoint $T$, such that $T^*=T$, where $T^*$ is the adjoint. I then have that the definition of a skew-...
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### Finding the rigid body transformation - how many points are required?

Apologies for a perhaps very stupid question, but I've begun to confuse myself a little, I think. If I have two sets of 3D points which are related by a transformation matrix, how many corresponding ...
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I am revising adjoints for a linear algebra exam and am confused as to how to prove this. Suppose that $T: V \rightarrow V$ has the property that $T^*=aT$ for some complex a. How then do you prove ...
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### Kernel of a polynomial which divides the minimal polynomial

Question: $P_{T}$:= characteristic polynomial of T; $minP_T$:= minimal polynomial of T. Consider $P_{T}=minP_T$. If $g$ is a polynomial which divides $minP_T$, then dim(Ker($g(T)$))=deg $g$. My ...
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### PMA Rudin: understanding the definition of matrices, definition 9.9

I have background of linear algebra but am still confused about the definition. Suppose $\{\mathbf{x_1}, \cdots, \mathbf{x_n}\}$ and $\{\mathbf{y_1}, \cdots, \mathbf{y_m}\}$ are bases of vector ...
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### Prove {$I$ , $A$ , $A^2$ ,…, $A^{n-1}$} is a basis for W

Let $F$ be a field , $A$ $\in$ $M_{n\times n}$($F$) (the set of all $n\times n$ matrices over field $F$ ) and $W$ = { $B \in M_{n\times n } (F)$ $|$ $AB = BA$ }. Suppose there exists a column ...
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### I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces

I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$. The linear ...
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### If each complex operator has invariant subspaces of any dimension?

I know that each operator in a $n$-dimensional complex space $A:E\to E$ is traingularizble. That is there exists a basis of the eigenvectors of $A$ wit which $A$ is triangular. Now this claim would ...
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### If for linear operator $A:E\to E$, $A^k=0$, then $A^n=0$ [duplicate]

Let for the linear operator $A:E\to E$, $A^k=0$, then $A^n=0$. Here $n=dim E$ and $n <k.$ I tried to find the characteristic oolynomial that I think it is the clue, however did nit succeed. Any ...
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### If $L: V\to V$ is a linear operator, and $V$ has bases of $n$ vectors, can we say rank$L = n$?

If $L: V\to V$ is a linear mapping, and $\{v_1, ..., v_n\}$ and $\{L(v_1), ..., L(v_n)\}$ are bases for $V$, can we say that rank$L = n$ ? I'm trying to prove that nullity$(L) = 0$, so my thought ...
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### Are all isometries of subsets affine?

I've found at least two questions that deal with whether isometries are affine, Are isometries always linear? and Should isometries be linear? However, both of these questions assume we are ...
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### Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
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### If there any plane curve whose critical points' curvature are invariant by linear transformation?

I was studying if the curvature of $f$: $$f(x) = \frac{ax}{b+x}$$ can have the critical points located at the same vertical than this other $g$ curve: $$g(x) = \frac{ax}{b+x} + cx = f(x) + cx$$ ...
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### Matrix representation of Fractional Linear Transformation and the Identity Matrix?

For $x \in \mathbb{R}$, define the fractional linear transformation of $x$ as $f(x)$ where: $$f(x) = \frac{ax + b}{cx+d}$$ Then $f(x)$ has a matrix representation in $\mathbb{R}^2$ of $F$ where: ...
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### Is there an easy way to remove scale from a squared linear transformation matrix

Given a linear transformation matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$, I know that one can use SVD or QR decomposition ...
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### Coordinate transformation in ODE's with a unit step function

Consider the following general set of 2 ODE's $$\dot{x}=\Theta(\dot{x} )f_1(x,y)+(1-\Theta(\dot{x}))f_2(x,y)$$ $$\dot{y}=(1-\Theta(\dot{y}))g_1(x,y)+\Theta(\dot{y})g_2(x,y)$$ where $\Theta(x)$ is ...
Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers. If $AB=BA$, prove that there exists a basis in which ...