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Exercise 2.1.22

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Let $T (1, 0, 0) = a$ , $T (0, 1, 0) = b$ , and $T (0, 0, 1) = c$ . Then we have

T (x, y, z) = xT (1, 0, 0) + yT (0, 1, 0) + zT (0, 0, 1) = ax + by + cz .

On the other hand, we have $T (x_{1}, x_{2}, \dots, x_{n}) = a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n}$ if $T$ is a mapping from $𝔽^{n}$ to $𝔽$ . To prove this, just set $T (e_{i}) = a_{i}$ , where ${e_{i}}$ is the standard of $𝔽^{n}$ .

For the case that $T : 𝔽^{n} \to 𝔽$ , actually we have

T (x_{1}, x_{2}, \dots, x_{n}) = (\sum_{j = 1}^{n} a_{1 j} x_{j}, \sum_{j = 2}^{n} a_{2 j} x_{j}, \dots, \sum_{j = m}^{n} a_{mj} x_{j}) .

To prove this, we may set $T (e_{j}) = (a_{1 j}, a_{2 j}, \dots, a_{mj})$ .

jephianlin

2011-06-27 00:00

Exercise 2.1.22

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