# 13.2 verify that ....... is a basis for R^2 find [v]_beta

• MHB
• karush
Linear Algebra" by David C. Lay.In summary, we are verifying that the given set is a basis for $\Bbb{R}^2$ and then finding the coordinates of a vector in that basis. The notation $[v]_\beta$ denotes the coordinates of vector $v$ in basis $\beta$, as taken from the textbook "Linear Algebra" by David C. Lay.
karush
Gold Member
MHB
Verify that
$\beta=\left\{\begin{bmatrix} 0\\2 \end{bmatrix} ,\begin{bmatrix} 3\\1 \end{bmatrix}\right\}$
is a basis for $\Bbb{R}^2$
Then for $v=\left[ \begin{array}{c}6\\8\end{array} \right]$, find $[v]_\beta$
ok, I presume next is
$c_1\begin{bmatrix} 0\\2 \end{bmatrix} +c_2\begin{bmatrix} 3\\1 \end{bmatrix}= \left[ \begin{array}{c}6\\8\end{array} \right]$
by augmented matrix we get (the book did this?)
$\left[ \begin{array}{cc|c} 0 & 3 & 6 \\ 2 & 1 & 8 \end{array} \right] =\left[ \begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 1 & 2 \end{array} \right]$
hence
$[v]_{\beta}=\left[ \begin{array}{c}3\\2\end{array} \right]$
following an example I don't think I understand the notation of $[v]_{\beta}$

$[v]_\beta$ denotes the coordinates of vector $v$ in basis $\beta$.

Could you say from which textbook this notation is taken?

https://www.physicsforums.com/attachments/9053
this one

## 1. What does it mean to "verify that [v]_beta is a basis for R^2?"

To verify that [v]_beta is a basis for R^2 means to check if the set of vectors [v]_beta spans the entire two-dimensional vector space of R^2 and is also linearly independent.

## 2. How do you determine if a set of vectors is a basis for R^2?

A set of vectors is a basis for R^2 if it satisfies the two conditions of spanning the entire vector space and being linearly independent. This can be checked by finding the determinant of the matrix formed by the vectors. If the determinant is non-zero, the vectors are linearly independent and thus form a basis for R^2.

## 3. What is the purpose of finding [v]_beta in this context?

The purpose of finding [v]_beta is to determine if the set of vectors is a basis for R^2. This is important in linear algebra as it helps in solving systems of linear equations and understanding vector spaces.

## 4. How is [v]_beta related to the beta coefficient in linear regression?

[v]_beta is not directly related to the beta coefficient in linear regression. The beta coefficient represents the change in the dependent variable for a unit change in the independent variable. [v]_beta, on the other hand, is a set of vectors used to determine if a set of vectors is a basis for R^2.

## 5. Can [v]_beta be any set of vectors to form a basis for R^2?

No, [v]_beta must satisfy the conditions of spanning the vector space and being linearly independent to form a basis for R^2. It is possible to have multiple sets of vectors that can form a basis for R^2, but they must all satisfy these conditions.

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