Exercise 3: Assume $A\in L(X,Y)$ and $A\v x=\v0$ only when $\v x=\v0$. Prove that $A$ is then 1-1.

Suppose $A(\v x_1)=A(\v x_2)$. Then \[ A(\v x_1-\v x_2)=A(\v x_1)-A(\v x_2)=\v0 \text{ implies that }\v x_1-\v x_2=\v0 \] so that $\v x_1=\v x_2$. Hence $A$ is 1-1.

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Principles of Mathematical Analysis

Exercise 9.3

Exercise 3: Assume $A \in L (X, Y)$ and $A \overset{ˇ}{x} = \overset{ˇ}{0}$ only when $\overset{ˇ}{x} = \overset{ˇ}{0}$ . Prove that $A$ is then 1-1.

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Suppose $A ({\overset{ˇ}{x}}_{1}) = A ({\overset{ˇ}{x}}_{2})$ . Then

A ({\overset{ˇ}{x}}_{1} - {\overset{ˇ}{x}}_{2}) = A ({\overset{ˇ}{x}}_{1}) - A ({\overset{ˇ}{x}}_{2}) = \overset{ˇ}{0} implies that {\overset{ˇ}{x}}_{1} - {\overset{ˇ}{x}}_{2} = \overset{ˇ}{0}

so that ${\overset{ˇ}{x}}_{1} = {\overset{ˇ}{x}}_{2}$ . Hence $A$ is 1-1.

analambanomenos

2023-08-07 00:00

Exercise 9.3

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