diff --git a/source/linear-algebra/source/02-EV/05.ptx b/source/linear-algebra/source/02-EV/05.ptx index 8604e89d3..3b8a8d088 100644 --- a/source/linear-algebra/source/02-EV/05.ptx +++ b/source/linear-algebra/source/02-EV/05.ptx @@ -268,6 +268,7 @@ In particular, the standard basis for \mathbb R^3 is \{\vec e_1,\vec e

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\left\{ @@ -300,8 +301,24 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea

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A.

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All pivot rows and columns: \left[\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 1 +\end{array}\right] \sim \left[\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 1 +\end{array}\right]

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\left\{ @@ -334,8 +351,24 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea

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D.

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Both a non-pivot row and non-pivot column: \left[\begin{array}{cccc} +2 & 2 & 4 & -3 \\ +3 & 0 & 3 & 0 \\ +0 & 0 & 0 & 1 \\ +-1 & 3 & 2 & 3 +\end{array}\right] \sim \left[\begin{array}{cccc} +1 & 0 & 1 & 0 \\ +0 & 1 & 1 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 0 & 0 +\end{array}\right]

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\left\{ @@ -369,8 +402,24 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea

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D.

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Both a non-pivot row and non-pivot column\left[\begin{array}{rrrrr} +2 & 2 & 3 & -1 & 4 \\ +3 & 0 & 13 & 10 & 3 \\ +0 & 0 & 7 & 7 & 0 \\ +-1 & 3 & 16 & 14 & 2 +\end{array}\right] \sim \left[\begin{array}{rrrrr} +1 & 0 & 0 & -1 & 1 \\ +0 & 1 & 0 & -1 & 1 \\ +0 & 0 & 1 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 +\end{array}\right]

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\left\{ @@ -403,8 +452,24 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea

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A.

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All pivot rows and columns: \left[\begin{array}{cccc} +2 & 4 & -3 & 3 \\ +3 & 3 & 0 & 6 \\ +0 & 0 & 1 & 1 \\ +-1 & 2 & 3 & 5 +\end{array}\right] \sim \left[\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 1 +\end{array}\right]

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\left\{ @@ -436,6 +501,21 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea

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C.

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All pivot columns, but a non-pivot row: \left[\begin{array}{rrr} +5 & -2 & 4 \\ +3 & 1 & 5 \\ +0 & 0 & 1 \\ +-1 & 3 & 3 +\end{array}\right] \sim \left[\begin{array}{rrr} +1 & 0 & 0 \\ +0 & 1 & 0 \\ +0 & 0 & 1 \\ +0 & 0 & 0 +\end{array}\right]

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