06-240/Classnotes For Thursday November 9: Difference between revisions
(Uploaded Week 9 Lecture 2 notes and Tutorial notes) |
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==Review of Last Class== |
==Review of Last Class== |
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But the matrix we now have represents a linear transformation <math>S</math> satisfying <math>S(v_1,\,v_2,\,v_3,\,v_4\,v_5)=(w_1,\,w_2,\,w_3,\,0,\,0)</math> for some bases <math>(v_i)_{i=1}^5</math> of <math>V</math> and <math>(w_j)_{j=1}^4</math> of <math>W</math>. Thus the image (range) of <math>S</math> is spanned by <math>\{w_1,w_2,w_3\}</math>, and as these are independent, they form a basis of the image. Thus the rank of <math>S</math> is <math>3</math>. Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of <math>T</math> must also be <math>3</math>. |
But the matrix we now have represents a linear transformation <math>S</math> satisfying <math>S(v_1,\,v_2,\,v_3,\,v_4\,v_5)=(w_1,\,w_2,\,w_3,\,0,\,0)</math> for some bases <math>(v_i)_{i=1}^5</math> of <math>V</math> and <math>(w_j)_{j=1}^4</math> of <math>W</math>. Thus the image (range) of <math>S</math> is spanned by <math>\{w_1,w_2,w_3\}</math>, and as these are independent, they form a basis of the image. Thus the rank of <math>S</math> is <math>3</math>. Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of <math>T</math> must also be <math>3</math>. |
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==Class Notes== |
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Revision as of 19:00, 15 November 2006
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Review of Last Class
| Problem. Find the rank (the dimension of the image) of a linear transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} whose matrix representation is the matrix A shown on the right. | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}} . |
| Theorem 1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T:V\to W} is a linear transformation and and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q:W\to W} are invertible linear transformations, then the rank of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the same as the rank of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle QTP} . | Proof. Owed. | |
Theorem 2. The following row/column operations can be applied to a matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
by multiplying it on the left/right (respectively) by certain invertible "elementary matrices":
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Proof. Semi-owed. |
Solution of the problem. using these (invertible!) row/column operations we aim to bring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to look as close as possible to an identity matrix, hoping it will be easy to determine the rank of the matrix we get at the end:
| Do | Get | Do | Get |
| 1. Bring a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/4} . | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}} | 2. Add times the first row to the third row, in order to cancel the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} in position 3-1. | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&-6&-8&-6&2\\6&3&2&9&1\end{pmatrix}} |
| 3. Likewise add times the first row to the fourth row, in order to cancel the in position 4-1. | 4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling). | ||
| 5. Turn the 2-2 entry to a by multiplying the second row by . | 6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" at position 2-2. | ||
| 7. Using three column operations clean the second row except the pivot. | 8. Clean up the row and the column of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} in position 3-3 by first multiplying the third row by and then performing the appropriate row and column transformations. Notice that by pure luck, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} at position 4-5 of the matrix gets killed in action. | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\end{pmatrix}} |
But the matrix we now have represents a linear transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(v_1,\,v_2,\,v_3,\,v_4\,v_5)=(w_1,\,w_2,\,w_3,\,0,\,0)} for some bases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (v_i)_{i=1}^5} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (w_j)_{j=1}^4} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} . Thus the image (range) of is spanned by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{w_1,w_2,w_3\}} , and as these are independent, they form a basis of the image. Thus the rank of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} . Going backward through the "matrix reduction" process above and repeatedly using theorems 1 and 2, we find that the rank of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} must also be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} .
Class Notes
Scan of Week 9 Lecture 2 notes
