Question

which sets are a basis for the following vector subspace of P2 :
X={A e M22 : A [1] = [0] }
[2] [0]

A {[2 0] , [0 -1] , [0 0] , [0 0] } C {[2 -1] , [0 0] }
[0 0] [0 0] [2 0] [0 -1] [0 0] [2 -1]

B{[2 -1] } D { [2 -1] , [2 -1] }
[2 -1] [2 -1] [-2 1]

Answer 1

Answer on Question #44283 – Math - Linear Algebra

Problem.

which sets are a basis for the following vector subspace of P2 :

$$X = \{A \in M22 : A[1] = [0]\}$$

[2] [0]

A $\{[2\ 0], [0\ -1], [0\ 0], [0\ 0]\} \subset \{[2\ -1], [0\ 0]\}$

[0 0] [0 0] [2 0] [0 -1] [0 0] [2 -1]

B$\{[2\ -1]\} \text{ D } \{[2\ -1], [2\ -1]\}$

[2 -1] [2 -1] [-2 1]

**Remark.** I suppose that the correct statement is

"Which sets are a basis for the following vector subspace of $P_2$: $X = \left\{A \in M_{22} : A\left[\begin{array}{c}1 \\ 2\end{array}\right] = \left[\begin{array}{c}0 \\ 0\end{array}\right]\right\}$

A $\left\{\begin{bmatrix}2 & 0 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & -1 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 2 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0 & -1\end{bmatrix}\right\}$

B $\left\{\begin{bmatrix}2 & -1 \\ 2 & -1\end{bmatrix}\right\}$

C $\left\{\begin{bmatrix}2 & -1 \\ 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 \\ 2 & -1\end{bmatrix}\right\}$

D $\left\{\begin{bmatrix}2 & -1 \\ 2 & -1\end{bmatrix}, \begin{bmatrix}2 & -1 \\ -2 & 1\end{bmatrix}\right\}$

Solution.

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ or $a + 2b = 0$ and $c + 2d = 0$, hence

$$a = -2b, \quad c = -2d.$$

Therefore

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} -2b & b \\ -2d & d \end{bmatrix} = -b \begin{bmatrix} 2 & -1 \\ 0 & 0 \end{bmatrix} -d \begin{bmatrix} 0 & 0 \\ 2 & -1 \end{bmatrix}.$$

So the basis for the given vector subspace is $\begin{bmatrix} 2 & -1 \\ 0 & 0 \end{bmatrix}, 2 \begin{bmatrix} 0 & 0 \\ 2 & -1 \end{bmatrix}$.

The correct answer is C.

**Answer:** C.

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