Question

Select all choices that are subspaces of \(\R^3\).
Note : \(\R\) denotes the set of real numbers.

• A. \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:x=\alpha\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}+\beta\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\alpha,\beta \in \R \right\}\)
• B. \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:x=\alpha^2\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}+\beta^2\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\alpha,\beta \in \R \right\}\)
• C. \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3=0,4x_1-2x_2+3x_3=0\right\}\)
• D. \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3+4-0\right\}\)

Answer 1

The Correct Option is A, C

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) : \(\left\{x=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \R^3:5x_1+2x_3=0,4x_1-2x_2+3x_3=0\right\}\)