• Title: Linear Independence (Definition)

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 22 | Linear Independence (Definition)

  • Bright video: https://youtu.be/UeHqH1yhoqU

  • Dark video: https://youtu.be/q66L7aw1QyE

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: la22_sub_eng.srt missing

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  • Quiz Content

    Q1: What is the correct definition for linear independence of a family $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ of vectors in $\mathbb{R}^n$?

    A1: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 = 0$.

    A2: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

    A3: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination for another vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

    A4: If $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.

    A5: If $\lambda_1 = \lambda_2 = \lambda_3 = 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.

    Q2: What is the correct definition for linear dependence of a family $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ of vectors in $\mathbb{R}^n$?

    A1: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where $\lambda_1, \lambda_2$ and $\lambda_3$ are not all zero.

    A2: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

    A3: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where we have $\lambda_1 = \lambda_2 = \lambda_3 = 0$.

    A4: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

    A5: If $\lambda_1 = \lambda_2 = \lambda_3 = 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.

  • Last update: 2024-10

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