46 lines
No EOL
1.1 KiB
TeX
46 lines
No EOL
1.1 KiB
TeX
\parindent=0pt
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From Unit 5, the following statements are equivalent.
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1. $A$ is invertible.
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2. $A^\tau$ is invertible.
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3. $Ax=b$ has a unique solution for any $b\in R^n$.
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4. $Ax=0$ has only the trivial solution.
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5. $A$ is row equivalent to $I_n$.
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6. $A$ is row equivalent to an invertible matrix.
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7. $A$ can be written as the product of elementary matrices.
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8. $A$ is not row equivalent to a matrix whose first row consists
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entirely of $0$.
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\bigskip
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{\it Trivia: $A$ is noninvertible iff it is row equivalent to a matrix
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with one of its rows consisting entirely of zero.}
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\bigskip
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From Unit 12, page 6.
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1. $T$ is one-to-one if and only if the nullity of $T$ is zero.
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2. $T$ is onto if and only if the rank of $T$ is the dimension of $W$.
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\bigskip
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From Unit 13, an inner product has the following properties.
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1. $\langle a,b\rangle=\langle b,a\rangle$
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2. $\langle a+b,c\rangle=\langle a,c\rangle+\langle b,c\rangle$
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3. $\langle r\cdot a,b\rangle=r\,\langle a,b\rangle$
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4. $\langle a,b\rangle\ge0$
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5. $\langle a,a\rangle=0$ if and only if $a=0$
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\end |