Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Computing bases for direct, wedge, tensor products, etc., of given vector spaces

I am filled with all kinds of vector space and I want to make sure I understand the basis for each kind of vector space. Suppose $\{v_i\}_{i=1}^n$ is the basis for vector space $V$, $\{w_j\}_{j=1}^m$ ...
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vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
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Integer-valued polynomial question

Let us have an $f(x)$ Integer-valued polynomial, which gains the value $1$ in $4$ different places. Prove, that in that case, it can't gain the value $-1$ on integer places. I tried with $f(x)-1=0$, ...
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Is the transformation $T: (r, \theta) \to (r, \theta + \phi)$ linear? Here $\phi$ is a given angle

Let $T$ rotate every point through the same angle $\phi$ about the origin, $i.e.$ $T: (r, \theta) \to (r, \theta + \phi)$ where $\phi$ is given. If in addition that $T(O) = O,$ namely, if $T$ maps the ...
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Real and imaginary part of an Eigenvector.

Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) generated by a band limited signal on a set of sensors. Is there a relation ...
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Finding shortest distance on earth

This is a math project for my linear algebra class. I have been having troubles figuring out if my answers are correct. I am using the dot product to figure out the great circle distance between two ...
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Terminology with linear transformation

I am working on a problem that asks me to "Write C for the matrix whose ij entry is $(1/2)^{ij}$" given that $M$ is the vector space of all $n x n$ matrices and $l$ is a linear transformation on $M$. ...
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What disqualifies an expression from being linear?

I'm taking some advanced math classes at my high-school, and I have some questions. A helpful answer would include a definition. Examples of a concept or definition given (no matter how simple; ...
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Understanding Jordan Canonical Form.

Two questions: How does the nilpotent index $k$ of a linear transformation L on a vector space of dimension $n$ relate to possible Jordan Canonical Forms? My understanding is that a Jordan block ...
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The matrix P is the transition matrix from what basis B to the basis B'

The Matrix $$P = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 1 \end{bmatrix}$$ is the transition matrix from what basis B to the basis B' = {(...
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Gram-Schmidt Process derivation question

I have a problem with this derivation I hope you can help me with: so we have constructed $w_{m+1}$ such that $g(w_{m+1},w_{m+1}) = 1$ and 0 otherwise, and we have show that $w_{m+1}$ is well defined,...
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Proof about linear functionals without reference to a basis.

Recall the following: Let $V$ be a vector space, and let $v$ be in $V$. If $\lambda(v) = 0$ for all $\lambda \in V^*$, then $v = 0$. Does anyone know of a proof that makes no reference to a basis? ...
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How do I Solve this Seemingly Simple Set of Four Equations with Four Unknowns?

I have what looks like a set of simple simultaneous equations: 4 equations with 4 unknowns. The numbers are really simple, and in fact I already know the answer, but I cannot figure out how to work ...
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Relation between $y=Ax_1$ and $y=WAx_2$

I have a question. Is there any relation between the following linear equations? $$y=Ax_1 \ \ \text{ and} \ \ y=WAx_2$$ W is diagonal square invertable matrix, A is an mxn matrix with $n>m$. I ...
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What happens when you add linear operators or take one linear operator inside another?

Let $A$ and $B$ be linear operators on $\mathbb R^2$. $A$ is the projection operator on the x-axis and $B$ is the the counterclockwise rotation by angle $\frac{\pi}{6}$. Find matrices of linear ...
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Eigen values of the operator $T : V \rightarrow V : T(f(t)) = t f~'(t)$
Let $V$ be the linear space of all real functions differentiable on $(0,1)$. If $f \in V,$ define $q = T(f(t))$ to mean that $q(t) = tf~'(t) ~\forall ~t \in (0,1)$ Prove that every real $\lambda$ is ...