Question:
Select all choices that are subspaces of \(\R^3\).
Note : \(\R\) denotes the set of real numbers.
Select all choices that are subspaces of \(\R^3\).
Note : \(\R\) denotes the set of real numbers.
Note : \(\R\) denotes the set of real numbers.
Updated On: Jan 30, 2026
- \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:x=\alpha\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}+\beta\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\alpha,\beta \in \R \right\}\)
- \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:x=\alpha^2\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}+\beta^2\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\alpha,\beta \in \R \right\}\)
- \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3=0,4x_1-2x_2+3x_3=0\right\}\)
- \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3+4-0\right\}\)
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The Correct Option is A, C
Solution and Explanation
The correct option is (A) : \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:x=\alpha\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}+\beta\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\alpha,\beta \in \R \right\}\) and (C) : \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3=0,4x_1-2x_2+3x_3=0\right\}\)
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