Show that u1, u2, u3 form a basis of R3 by proving that they are independent and Span{u1, u2, u3}= R3. Given the following three vectors:\[ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \]Show that $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ form a basis of $\mathbb{R}^3$ by proving that they are independent and $\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3$. To justify $\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3$, show that for a random vector:\[\mathbf{y} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3, \quad \mathbf{y} \in \text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \, \text{or} \, \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \alpha \mathbf{v}_1 + \beta \mathbf{v}_2 + \gamma \mathbf{v}_3, \, \alpha, \beta, \gamma \in \mathbb{R}\]

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Given the following three vectors: Show that v1, v2, iz form a basis of R by proving that they are independent AND Span {di, ö2, üz} = R*. To justify Span {v1, d2, vz} = R°, show that for a random vector y e R°, ye Span {v1, 02, 03} , or = a•-0j+ß-7z+y•7g, a, ß, y E R

Given the following three vectors: Show that v1, v2, iz form a basis of R by proving that they are independent AND Span {di, ö2, üz} = R*. To justify Span {v1, d2, vz} = R°, show that for a random vector y e R°, ye Span {v1, 02, 03} , or = a•-0j+ß-7z+y•7g, a, ß, y E R

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

Show that u1, u2, u3 form a basis of R3 by proving that they are independent and Span{u1, u2, u3}= R3.

Given the following three vectors:

\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}
\]

Show that $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ form a basis of $\mathbb{R}^3$ by proving that they are independent and $\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3$.

To justify $\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3$, show that for a random vector:

\[
\mathbf{y} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3, \quad \mathbf{y} \in \text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \, \text{or} \, \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \alpha \mathbf{v}_1 + \beta \mathbf{v}_2 + \gamma \mathbf{v}_3, \, \alpha, \beta, \gamma \in \mathbb{R}
\]

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