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M = \begin{bmatrix} a & b \\ c & 0 \\ \end{bmatrix}

with $a + b - 2c = 0$

Show that M is a subspace from $M_{2:2} (\mathbb{R})$

[$M_{2:2} (\mathbb{R})$ is the ring of 2 \times 2 matrices over the real numbers]

can someone help me?

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To show $M$ is a subspace, you need to show that $M$ is closed under addition and scalar multiplication.

Scalar Multiplication: $$k\begin{bmatrix} a & b \\ c & 0 \\ \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & 0 \\ \end{bmatrix}$$ Since we know $a + b - 2c = 0$, what can we say about $ka + kb - k2c$?

Addition: $$\begin{bmatrix} a & b \\ c & 0 \\ \end{bmatrix} + \begin{bmatrix} d & e \\ f & 0 \\ \end{bmatrix} = \begin{bmatrix} a +d & b+e \\ c+f & 0 \\ \end{bmatrix}$$ This time we know that $a + b - 2c = 0$ and $d + e - 2f = 0$. So what can we say about $(a+d) + (b +e) - 2(c + f)$?

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thanks for Deven Ware... your answer help me... – Zuhair Oct 31 '12 at 0:47

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