# Questions tagged [change-of-basis]

This tag is for question about changing basis of a finite dimensional vector space. For example, how does the representation of a vector, or a matrix change with the change of basis. Please don't use this tag on its own, it is better to add a more general tag which is relevant to your question, e.g. [linear-algebra] or [matrices] for better visibility.

521 questions
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### Find basis for $\Bbb Q(\sqrt 2 + \sqrt2)$ over $\Bbb Q(\sqrt 2)$ [on hold]

I need to find the basis for this which is apparently $\{\sqrt 2, (\sqrt 2)^2, 1\}$. Thanks so much for your help
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### How many bases can be formed given a set of vectors? [duplicate]

Let $v_1,\cdots,v_n$ be a set of vectors such that the generated vector space has dimension $r$. How many different bases with dimension $r$ can be formed from these vectors? Don't understand the ...
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### Understanding diagonalization through change of basis

I'm sorry if what I am asking is unclear, but I was hoping that somebody might be able to help me understand diagonalization a bit more by explaining it through the lens of a change of basis. I ...
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### Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? $P$ is the change of basis matrix from $B$ to $C$ and $T$ is a linear transformation

Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? ($P$ is the change of basis matrix from $B$ to $C$ (both vector spaces) and $T$ is a linear transformation). It would make a lot of ...
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### Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B =$ {$1+x, 1-x, x^2$}

Finding the matrix of $T(p(x)) = p(2x-1)$ with respect to the basis $B =$ {$1+x, 1-x, x^2$} To find the matix of a transformation with respect to a given basis, I find the images of the basis ...
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### Basis functions and weak ODE solution

Given some linear differential operator $L$, I'm trying to solve the eigenvalue problem $L(u) = \lambda u$. Given basis functions, call them $\phi_i$, I use a variational procedure and the Ritz method ...
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### How do you convert $6.75$, in base $10$, to base $2$?

Can I get a detailed explanation on how to convert $6.75$ base $10$ to $2$. I have really tried all I can but I don't still get it.
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### Why does row reducing an augmented matrix of two basis written in the coordinates of a third give the change of basis matrix?

Sorry for the longwinded and vague title. I've included a full picture of the theory (taken from my phenomenal textbook, David Poole's Linear Algebra: A Modern Introduction-Brooks Cole (2014)) below. ...
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### Matrix of endomorphism in different basis

Let $E$ be a vector space, with $\dim E = n$. Let $u$ be an endomorphism of $E$ and $B = (b_1, \dots, b_n)$, $C = (c_1, \dots, c_n)$ be two basis of $E$. Let $\operatorname{Mat}_{BC}(u)$ denote the ...
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### How can I get transformed coordinate between different basis?

For example, there is set of DCT basis which are orthogonal: $F_1(x,y),F_2(x,y)\cdots,F_N(x,y)$. So, given function can be uniquely expressed as sum of DCT basis. (we decompose an image to sum of DCT ...
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### Matrix representation of a linear transformation under change of basis.

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$ such that $x\mapsto Ax$. Suppose A is diagonalizable and $B=\{v_1,...,v_n\}$ is a ...
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### Change of basis…

**Question:**Let $(1, z, z^2)$ be the standard basis for $\mathbb{C}^2$. Let $(1, z - 1, (z - 1)^2$ be another basis for $\mathbb{C}^2$. Find a matrix transformation from $p(z) = a_0 + a_1z + a_2z^2$ ...
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### What is the general method of expressing a basis in terms of another basis of a vector space?

I want to understand basis of vector spaces more clearly. I know that the basis of a vector space is a set of linearly independent elements that span that vector space. I know that an element in a ...
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### Change of base in $R^2$

In the official solution of an university exam I see the following question: Given the base vectors $$B = \{ \begin{pmatrix} -1 \\ 1 \end{pmatrix} , \begin{pmatrix} 1 \\ 1 \end{pmatrix} \}$$ and ...
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### Finding linear transformation from matrix

I tried solving this question by taking given matrix $[A]_{B_{1}}$ = $[T]_{B_{2}}$. The answer that I am getting is $(35,-10)$. Tried solving several times and still reaching at the same answer. ...
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### Prove that there exists a semi-orthogonal $U$ such that $U^TAU=B$, where $A$ and $B$ are positive-definite symmetric matrices.

Let there be a semi-orthogonal matrix $U \in \mathbb{R}^{m\times n}$ such that $U^TU=I_n$ if $m > n$ If $A \in \mathbb{R}^{m\times m}$ and $B \in \mathbb{R}^{n\times n}$ are positive-definite ...
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### Is a Change of Basis Matrix equivalent to the matrix inverse in this case?

I was looking at how to construct a change of basis so that when given a system of linear equations one could change the associated matrix into a diagonal matrix -- thus making the system easier to ...
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### Intuition for the invariance of the determinant under change of basis

$$A' = PAP^{-1}$$ $$\det(A')=\det(P)\det(A)\det(P^{-1})=\det(A)$$ Now, that makes sense algebraically, but consider the below diagram: This a geometric representation of the two 'normal' basis ...
$$B=\left\{\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\right\} \quad C=\left\{\begin{bmatrix} 4 \\ 5 \\ 6 \end{... 1answer 29 views ### Let (v_1, v_2) be a basis for \text{Im}(p) and (v_3) be a basis for \ker(p). Prove that B' = (v_1, v_2, v_3) is a basis for V . Let B = (1, X, X^2) be a basis for \mathbb{R}_2[X] and p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big) be the linear map defined by p(1) = \frac{1}{3}(2 − X − X^2), p(X) = \frac{1}{3}(−1 + 2X − X^2)... 2answers 20 views ### How to do the following change of Basis V=\mathbb{Q}^3 The two following Bases of V are given S=\{(1,0,0) , (0,1,0) , (0,0,1)\}, T=\{(-4,2,1) , (-1,0,1), (1,-1,1)\}; Let w ∈ V with γ_T$$(w)$=$ $$\begin{pmatrix}1\\0\\1\end{pmatrix}... 1answer 18 views ### Two Change of Basis Matrices Let V = P_2 and dim(v) = 3. Let B = {1, x, x^2} and S = {1+x, 2-x, 3+x^2} be two basis of V. What is the matrix P_B,_S and P_S,_B? I am not entirely sure how to ... 1answer 21 views ### Compute \text{rk}(p) using Gauss reduction on A. Compute \dim\big(\ker(p)\big). Let B = (1, X, X^2) be a basis for \mathbb{R}_2[X] and p ∈ \mathcal{L}\big(\mathbb{R}_2[X]\big) be the linear map defined by p(1) = \frac{1}{3}(2 − X − X^2), p(X) = \frac{1}{3}(−1 + 2X − X^2)... 1answer 43 views ### Prove that p ◦ p = p. (Representing a Linear Transformation as a Matrix) Let B = (1, X, X^2) be an ordered basis for \mathbb{R}_2[X] and p ∈ \mathcal{L}(\mathbb{R}_2[X]) be the linear map defined by p(1) = \frac{1}{3}(2 − X − X^2), p(X) = \frac{1}{3}(−1 + 2X − X^2)... 2answers 35 views ### Transformation matrix with respect to an orthonormal basis I have this question here... Let V be the span of v_{1}=(0,1,2), v_{2}=(-1,0,1) and v_{3}=(-1,1,3). (a)Construct an orthonormal basis B' for V (usual dot product). (b) ... 0answers 69 views ### Christoffel symbol transformation The Christoffel symbols transform like$$\Gamma^{\prime a}_{bc} = \frac{\partial x^{\prime a}}{\partial x^d} \frac{\partial x^e}{\partial x^{\prime b}} \frac{\partial x^f}{\partial x^{\prime c}} \...
It's known that a B-spline of degree $p$ , $B_j^p(x)$ is completely determined by a knot vector $(u_j,u_{j+1},...,u_{j+p+1})$. One could define it as: \$B_j^p(x)=[u_j,u_{j+1},...,u_{j+p+1}](\cdot - x)^...