Tagged Questions

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Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
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Rotate with a transformation matrix while preserving shape

I used excel to plot a function using discrete points. I'd like the function to rotate about the origin by applying a rotation transform function to each pair of coordinates. However, I can't figure ...
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Prove that the following matrices cannot represent the linear transformation $T$ in ANY basis

$T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined as $T(x,y,z) = (2x,z,y)$ is a linear transformation. I need to prove that the following matrices cannot represent $T$ in ANY basis: ...
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Linear Algebra One to one and onto function

I was just wondering how I can tell if a function is onto. $\mathbf{R}^3\to\mathbf{R}^1$ Lets say the standard transformation matrix has vectors $\{1,0,0\}$, $\{0,1,0\}$, $\{0,0,0\}$. I know that this ...
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Linear transformation and linear subspaces

Let $T:V\rightarrow V$ bwe a linear transformation. Let $L \subset V$ be a linear subspace such that $L \cap \text{Ker}\,(T)=\{0 \}$. Prove that the image given by T of any linear independent ...
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Real linear tranformation

When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear? I need to know it for my homework but I can't seem to find the definition anywhere.
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Question on linear algebra mappings

If $T:R^m\to R^n$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M|h|$ for $h\in R^m$.
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when linear mapping keeps monotonicity of $L_2$ norm

Consider an arbitrary vector $\alpha$ from vector space $R^p$, a linear mapping $A: \alpha\rightarrow A\alpha$ transforms $\alpha$ to $A\alpha$ in space $R^q$. What condition should $A$ satisfy so ...
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finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...
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Givens rotation of the following vector of 3 elements.

I have to find the givens rotation matrix that will transform the following vector $[1, 1, -1]^T$ to $[y, 1, 0]^T$ (basically to insert a $0$ on the third position without altering the second one). I ...
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Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
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How to compute (and check) this transform matrix?

Background: This is a homework exercise which asks to compute a transform matrix. The answer has been published by our teacher. However, my approach goes a different way and gets a different solution. ...
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Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
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Prove or disprove that there exists a linear map given a set of vectors and their mapping

I'm stuck on this seemingly simple homework question, but I just don't know how to approach it at all :( Here is the question: " Prove or disprove that there is a Linear map ...
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How can I show that the kernel of $f-id_V$ is equal to the image of $f$?

How can I show that $\ker (f-id_V)=\Im f$ given that $f:V\longrightarrow V$ is a linear transformation such that $f\circ f=f$? Thank you.
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“Well defined” function - What does it mean?

What does it mean for a function to be well-defined? I encountered with this term in an excersice asking to check if a linear transformation is well-defined.
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Proving a subspace under a linear transformation by the closure of standard addition and scalar multiplication

$T(x,y,z)= (3x-2y, -2x+3y, 5z)$ be a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ Show that $A= \{(u,v,z) \in \mathbb{R}^3~|~(u,v,w)=T(x,y,z)\}$ for some $(x,y,z)$ in $\mathbb{R}^3$ is ...
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some corollaries of the rank - nullity theorem

Here is a problem which I encountered in linear algebra. I realized that it might be a corollary of the "rank - nullity theorem" but I don't know how to work with it. Hope you can help! Thank you! ...
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Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
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Let $A = (1,3) (2,5)$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ ...
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Linear Algebra Transformation Question

Suppose 3x3 matrix A = [122,011,012](commas separating rows) And suppose T: R^3 -> R^3 be defined by T(x) = A(x) for every x in R^3 (a) Find A^-1 (Easy) (b) Suppose T^-1 be the inverse transformation ...
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Linear transformations… $T(\mathbf v)\ne0$, $T^2 - 0$… Prove $\mathbf v$ and $T(\mathbf v)$ are linearlly independent…

Suppose $T:\mathbb R^n \to\mathbb R^n$ is a linear transformation and suppose that $\mathbf v$ is a vector such that $T(\mathbf v) \ne 0$ but $T^2(\mathbf v) = 0$ (where $T^2 = T \circ T$) Prove ...
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Find the matrix of $T$ given :$T([3, 5]) = [2, -1]$ and $T([1, 2]) = [3, 7]$ , $T$ is linear.

I know that T(v) = v' = Av , where v is the vector, v' is the image, and A is the matrix of transformation So I've set the two images (v') equal to the matrix [S sub1, S sub 2] What does [S sub1, S ...
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Rotate a triangle specified by vectors around its center

Well, I know that, in order to rotate a triangle specified by three vectors in $R^2$ we just rotate each vector in the same angle, and to do this we apply the rotation matrix in ...
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Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...
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Linear transformations and eigenvalues [duplicate]

Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same. Can anyone provide ...
if $Ax=b$ is consistent, then the solution set of $Ax=b$ is obtained by translating the solution set of $Ax=0$ is it true or false? or is it sometimes false and sometimes true?
Im trying to create matrix which rotates vector. I have $\vec{g}=(g_1,g_2,g_3);\:g_1\in\mathbb{R},g_2\in\mathbb{R},g_3\in\mathbb{R}$ - it represents gravitation. And $\vec{o}=(o_1,o_2,o_3)$ is vector ...