• Title: Coordinates with respect to a Basis

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 25 | Coordinates with respect to a Basis

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  • Quiz Content

    Q1: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

    A1: $\lambda_1 = 0$ and $\lambda_2 = 1$

    A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

    A3: $\lambda_1 = 1$ and $\lambda_2 = 1$

    A4: $\lambda_1 = 4$ and $\lambda_2 = 5$

    A5: $\lambda_1 = 0$ and $\lambda_2 = 3$

    Q2: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 3 \ 1 \ 3 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

    A1: $\lambda_1 = 1$ and $\lambda_2 = 1$

    A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

    A3: $\lambda_1 = 0$ and $\lambda_2 = 1$

    A4: $\lambda_1 = 4$ and $\lambda_2 = 5$

    A5: $\lambda_1 = 0$ and $\lambda_2 = 3$

    Q3: Consider the subspace $U \subseteq \mathbb{R}^3$ spanned by the basis $\mathcal{B} = \Big( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} \Big) $. What are the coordinates of the vector $\begin{pmatrix} 1 \ 5 \ 0 \end{pmatrix} $ with respect to the basis $\mathcal{B}$?

    A1: $\lambda_1 = 5$ and $\lambda_2 = 1$

    A2: $\lambda_1 = 2$ and $\lambda_2 = 1$

    A3: $\lambda_1 = 1$ and $\lambda_2 = 1$

    A4: $\lambda_1 = 1$ and $\lambda_2 = 5$

    A5: $\lambda_1 = 0$ and $\lambda_2 = 3$

  • Last update: 2024-10

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