Suppose v⃗1, v⃗2, …, v⃗m are arbitrary vectors in ℝ^n Consider the transformation from ℝ^m to ℝ^

Name: Problem 7, Chapter 2: Linear Algebra With Applications
Uploaded: 2023-02-05T22:47:44-08:00
Channel: Victor Salazar
Description: Suppose v⃗1, v⃗2, …, v⃗m are arbitrary vectors in ℝ^n Consider the transformation from ℝ^m to ℝ^n given by T[ x1 x2 ⋮ xm ]=x1v⃗1+x2v⃗2+⋯+xmv⃗m Is this transformation linear? If so, find its matrix A in terms of the vectors v⃗1, v⃗2, …, v⃗m

step 1 For this exercise, we have a multi -set problem. And we're going to start at the first step. We

Question

Suppose $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$ are arbitrary vectors in $\mathbb{R}^{n}$ Consider the transformation from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ given by \[ T\left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{array}\right]=x_{1} \vec{v}_{1}+x_{2} \vec{v}_{2}+\cdots+x_{m} \vec{v}_{m} \] Is this transformation linear? If so, find its matrix $A$ in terms of the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$

   Suppose $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$ are arbitrary vectors in $\mathbb{R}^{n}$ Consider the transformation from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ given by
\[
T\left[\begin{array}{c}
x_{1} \\
x_{2} \\
\vdots \\
x_{m}
\end{array}\right]=x_{1} \vec{v}_{1}+x_{2} \vec{v}_{2}+\cdots+x_{m} \vec{v}_{m}
\]
Is this transformation linear? If so, find its matrix $A$ in terms of the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$

Linear Algebra With Applications

Otto Bretscher 5th Edition

Chapter 2, Problem 7

Instant Answer

Solved by Expert Victor Salazar

02/05/2023

Step 1

A transformation is linear if it satisfies two properties: additivity and homogeneity. The transformation $T$ is additive if $T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})$ and it is homogeneous if $T(c\vec{x})=cT(\vec{x})$ for any scalar $c$. Show more…

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Suppose $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$ are arbitrary vectors in $\mathbb{R}^{n}$ Consider the transformation from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ given by \[ T\left[\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{array}\right]=x_{1} \vec{v}_{1}+x_{2} \vec{v}_{2}+\cdots+x_{m} \vec{v}_{m} \] Is this transformation linear? If so, find its matrix $A$ in terms of the vectors $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$

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Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. This means that for any vectors u and v and any scalar c, the transformation T satisfies T(u + v) = T(u) + T(v) and T(cu) = cT(u). The property ensures that the structure of the vector space is maintained under the transformation.

Matrix Representation of a Linear Transformation

Any linear transformation from one Euclidean space to another can be expressed as a matrix multiplication. When the inputs are expressed in terms of the standard basis, the output of the transformation is a linear combination of the images of these basis vectors. Consequently, the matrix representation consists of columns that are the images of the standard basis vectors, meaning that if each column is given by a vector from the transformation, the matrix A constructed by taking these vectors as columns will satisfy T(x) = A x for all x.

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