Questions tagged [change-of-basis]

Question 1

In $\mathbb{R}^2$, consider the following ordered basis: $$B=((1,0),(0,1)),\text{ }C=((-1,1),(1,1)),\text{ and }D=((\sqrt{3},1),(\sqrt{3},-1))$$ Find the change of basis matrix from $B$ to $C$, from $...

Question 2

For a linear transformation A and a k-blade $b=v_1\wedge v_2 \wedge\dots\wedge v_k$ we take the outermorphism of $A$ acting on $b$ to mean $\underline A(b):= A(v_1)\wedge A(v_2)\wedge\dots\wedge A(v_k)...

Question 3

This problem appears in the book: Linear Algebra and its applications - David C. Lay - Fourth Edition It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18 $(4.7), \...

Question 4

I am studying differentiable manifolds and I came across the definition of cotangent space. I have a doubt on how we change coordinates in the cotangent space. Let $(A,\varphi)$ and $(B,\psi)$ be ...

Question 5

The statement of Gram-Schmidt orthonormalisation thm is: Let $\{v_1, v_2, \dots, v_n\}$ be a linearly independent set of vectors in an inner product space $V$. Then there exists an orthogonal set of ...

Question 6

This question mainly based on Jnez71 answer How is the Laplace Transform a Change of basis? We are interested in a specific basis known as the "Fourier" basis. The Fourier basis can be ...

Question 7

Context: the question came up while discussing function approximations in the context of computer graphics. In particular, (truncated) spherical harmonics (SH) are a popular way to encode a spherical ...

Question 8

I'm having trouble with some conflict between my book of linear algebra and my professor notes on it. The book (Poole's Linear Algebra) stated the following: Let $\beta = \{\boldsymbol{u}_1, . . . , \...

Question 9

If $T:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear transformation and $\vec{e}_1,\dots,\vec{e}_n$ are the usual base vectors for $\mathbb{R}^n$ then $T(\vec{e}_1),\dots,T(\vec{e}_n)$ is a ...

Question 10

Suppose a real matrix $\mathbf{A}^{m\times n}$ with $m>n$, and $\text{rank}(\mathbf{A})=r<n$. Suppose now a real matrix $\mathbf{E}^{s\times n}$ of full rank, $s=n-r$, and in a such way that $$ \...

Question 11

I've just started working my way through Shahshahani’s An Introductory Course on Differentiable Manifolds, and there's something right at the beginning that I don't quite follow. When checking how ...

Question 12

Let $\mathbb{K}$ be a field, and let $P \in \mathbb{K}_n[X]$ be a polynomial of degree $n$. Let $a_0, \ldots, a_n \in \mathbb{K}$ be $n+1$ pairwise distinct scalars. Define the family $\mathcal{F}=\...

Question 13

Let $ V $ be a vector space with an inner product $ g $ and let $\{v_i\} $ be a basis of $ V $. Consider another basis $ \{u_i\} $ that is $ g$-orthonormal, and let $ B $ be the change-of-basis ...

Question 14

I'm having trouble with the following problem: I get that we have: $$f\left(e_1\right)=-e_2-e_3 \sin (v) \qquad f\left(e_2\right)=e_1+e_3 \cos(v)\qquad f\left(e_3\right)=-e_1 \sin (v)+e_2 \cos (v)$...

Question 15

The first course I read of Linear Algebra was French, here you can find it. In the "Matrices et applications linéaires">"Changement de bases" section the matrix that serves as ...

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Questions tagged [change-of-basis]

Finding change of basis matrices in $\mathbb{R}^2$ [closed]

How do i show that the outermorphism of a linear transformation is basis independent [closed]

Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra

Change of coordinates in the cotangent space

Why the second part of the statement is given in Gram-Schmidt orthonormalisation thm

How Fourier transform Viewed as Change of basis?

Does any nice finite-dimensional subspace of $L^2$ have an orthonormal basis of nonnegative functions?

Difference between change of basis matrix and change of coordinates matrix.

Obtain a basis for $\mathbb{R}^n$ by interchanging the image of one basis vector with $\vec{e}_1$

Orthogonal complement and relations between matrices formed from its basis

Transpose in change of basis

Can we prove the independence of the family $\left(P\left(X+a_i\right)\right)$ without using matrices?

Relationship Between Metric Determinant and Change-of-Basis Matrix

Trouble in problem about matrix of linear transformation after change of basis?

Is the "change of basis matrix" the same as "matrice to passage"? [duplicate]

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