# 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)

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### Consider the set $S = \{x : Ax = 0\}$. Show that $\dim (S) = n-r$ where $r$ is the number of Linearly Independent rows of $A$.

Consider the set $S = \{x : A_{m \times n}x = 0\}$ where $x \in \mathbb{R}^n$. Show that $\dim (S) = n-r$ where $r$ is the number of Linearly Independent rows of $A$. Can we directly use the rank-...
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### What's the difference between homogeneous coordinates and projected coordinates?

I don't get how people are using homogeneous coordinates in order to construct the projection of an object. I know that homogeneous coordinates allow us to perform affine transformations in higher ...
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### Rotation by an angle θ [closed]

How to find a 4x4 matrix that represents in homogeneous coordinates the rotation by an angle θ about the $p=t(1,1,1)^T+ (1,0,0)^T$ line of $R^3$ ? Thank you!
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### How do I find a corresponding point on a triangle that is projected onto a different triangle? [closed]

let's say I have a triangle with vertices: ...
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### Mapping a rectangle onto the unit square

How to find the matrix in homogeneous coordinates that maps the rectangle with vertices $(1,-2)$, $(1,2)$, $(4,2)$, $(4,-2)$ onto the unit square? I have really searched a lot but couldn't find the ...
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### Hamilton-Cayley Theorem for Vector Spaces Over Any Field

In Serge Lang’s third edition of Linear Algebra on p. 243, he gives the result of the Hamilton-Cayley theorem for linear operators for vector spaces over any field $K$. My problem is that I don’t ...
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### A counter for: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable.

Let $V$ be a finite dimensional vector space and $f:V \rightarrow V$ invertible linear transformation. Prove or disprove: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable. I know it's not ...
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### Proving a vector space cannot exist

Hamilton tried to find a $3$-dimensional number system with the following properties: Every number can be written by $a + bx + cy$. This means every real number $a$ can be represented by $a + 0x + 0y$...
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### Finding a flaw in a linear algebra proof

Consider $f(x, y, z) = (x^2 + 2y^2, x + z, x - z)$. Your friend wants to find the image of $f$, and their proof is the following: We know that $w$ is in the image of $f$ if and only if there is a ...
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### Finding the kernel of a linear transformation between polynomials

Find the kernel of the linear transformation of $T:P^4\rightarrow P^4$ defined by $$T(p) = p'' -p'-p$$ Previously I had to prove this was a linear transformation, and was successful, but I am having ...
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### Continuity of the linear map $(x_n)_{n \in \mathbb{N}} \mapsto \sum_{k=1}^{\infty}k\cdot x_k$

Let E:\{(x_n)_{n \in \mathbb{N}} : \forall k \ x_k \in \mathbb{R}, \exists N\ge 0: \forall n \ge N \ x_n = 0 \} \\ L:(E,\|\cdot \|_\infty)\rightarrow\mathbb{R},\ \ \ (x_n)_{n \in \mathbb{N}} \...