In Exercises 1-4, find the vector 𝐱 determined by the given coordinate vector [𝐱]ℬ and the given

Name: Problem 2, Chapter 4: Linear Algebra and Its Applications
Uploaded: 2019-08-12T23:41:37-08:00
Duration: 41 s
Channel: Doruk Isik
Description: In Exercises 1-4, find the vector 𝐱 determined by the given coordinate vector [𝐱]ℬ and the given basis ℬ . ℬ={[ 4 5 ],[ 6 7 ]},[𝐱]ℬ=[ 8 -5 ]

step 1 In this problem we're asked to find vector x using the given basis and the given coordinate vect

Question

In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$ $$ \mathcal{B}=\left\{\left[\begin{array}{l}{4} \\ {5}\end{array}\right],\left[\begin{array}{l}{6} \\ {7}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{8} \\ {-5}\end{array}\right] $$

   In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{l}{4} \\ {5}\end{array}\right],\left[\begin{array}{l}{6} \\ {7}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{8} \\ {-5}\end{array}\right]
$$

Linear Algebra and Its Applications

David C. Lay, Steven… 5th Edition

Chapter 4, Problem 2

Instant Answer

Solved by Expert Doruk Isik

08/12/2019

Step 1

In this case, the basis vectors are $\left[\begin{array}{l}{4} \\ {5}\end{array}\right]$ and $\left[\begin{array}{l}{6} \\ {7}\end{array}\right]$. We can denote these as $\mathbf{b}_1$ and $\mathbf{b}_2$ respectively. Show more…

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In Exercises $1-4,$ find the vector $\mathbf{x}$ determined by the given coordinate vector $[\mathbf{x}]_{\mathcal{B}}$ and the given basis $\mathcal{B} .$ $$ \mathcal{B}=\left\{\left[\begin{array}{l}{4} \\ {5}\end{array}\right],\left[\begin{array}{l}{6} \\ {7}\end{array}\right]\right\},[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{r}{8} \\ {-5}\end{array}\right] $$

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Key Concepts

Linear Combination

The process of forming a vector as a sum of scalar multiples of basis vectors is known as a linear combination. The scalars from the coordinate vector serve as the multipliers for each corresponding basis vector.

Basis

In any vector space, a basis is a set of linearly independent vectors that spans the entire space, meaning every vector in the space can be expressed as a unique linear combination of these basis vectors.

Coordination Vector Representation

A coordinate vector represents the coefficients used in the linear combination of the basis vectors. It indicates how much of each basis vector is needed to construct a given vector in the space.

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