14-240/Tutorial-November11 - Revision history

Bug at 22:35, 30 November 2014

2014-11-30T22:35:03Z

← Older revision		Revision as of 18:35, 30 November 2014
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	==Boris==		==Boris==

	====~~Background~~====		====Useful Definitions====

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.

Bug at 22:25, 30 November 2014

2014-11-30T22:25:36Z

← Older revision		Revision as of 18:25, 30 November 2014
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	'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f(x)) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>B, K</math>?		'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f(x)) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>B, K</math>?

			==Nikita==

Bug: /* Coordinate and Matrix Representation Problems */

2014-11-30T00:08:52Z

Coordinate and Matrix Representation Problems

← Older revision		Revision as of 20:08, 29 November 2014
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	==Boris==		==Boris==

			====Background====
	====Coordinate and Matrix Representation Problems====

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math>is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.


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	Boris's Problems:		====Boris's Problems====


	Let <math>S</math> be the standard ordered basis of <math>P_n(F)</math>.		Let <math>B</math> be the standard ordered basis of <math>P_n(F)</math> and <math>K</math> be the standard ordered basis of <math>F</math>.


	'''Q1'''. What is the '''coordinate vector''' of <math>x^2 + x^5</math> relative to <math>S</math>?		'''Q1'''. What is the '''coordinate vector''' of <math>x^2 + x^5</math> relative to <math>B</math>?


	'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>S</math>?		'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f(x)) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>B, K</math>?

Bug: /* Coordinate and Matrix Representation Problems */

2014-11-30T00:02:23Z

Coordinate and Matrix Representation Problems

← Older revision		Revision as of 20:02, 29 November 2014
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	====Coordinate and Matrix Representation Problems====		====Coordinate and Matrix Representation Problems====

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate ~~representation~~''' of <math>v</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate vector''' of <math>v</math> relative to <math>B</math>is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.


	Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math> and <math>K = \{v_1, v_2, v_3, ..., v_m\}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then <math>T(v_j) = \displaystyle\sum_{i=1}^{m} c_{ij}T(v_j)</math> where <math>c_{ij} \in F</math>. Then the '''matrix representation''' of <math>T</math> in the ordered bases <math>B, K</math> is the matrix <math>\begin{pmatrix}c_{11} & c_{12} & c_{13} & ... & c_{1n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\c_{31} & c_{32} & c_{33} & ... & c_{3n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\... & ... & ... & ...& ... \\c_{m1} & c_{m2} & c_{m3} & ... & c_{mn}\end{pmatrix}</math>.		Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math> and <math>K = \{v_1, v_2, v_3, ..., v_m\}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then <math>T(v_j) = \displaystyle\sum_{i=1}^{m} c_{ij}T(v_j)</math> where <math>c_{ij} \in F</math>. Then the '''matrix representation''' of <math>T</math> in the ordered bases <math>B, K</math> is the matrix <math>\begin{pmatrix}c_{11} & c_{12} & c_{13} & ... & c_{1n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\c_{31} & c_{32} & c_{33} & ... & c_{3n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\... & ... & ... & ...& ... \\c_{m1} & c_{m2} & c_{m3} & ... & c_{mn}\end{pmatrix}</math>.


			Boris's Problems:


			Let <math>S</math> be the standard ordered basis of <math>P_n(F)</math>.


			'''Q1'''. What is the '''coordinate vector''' of <math>x^2 + x^5</math> relative to <math>S</math>?


			'''Q2'''. Let <math>T:P_n \to F</math> be a linear transformation that is defined by <math>T(f) = f(0)</math>. What is the '''matrix representation''' of <math>T</math> in <math>S</math>?

Bug: /* Coordinate and Matrix Representation Problems */

2014-11-29T23:52:14Z

Coordinate and Matrix Representation Problems

← Older revision		Revision as of 19:52, 29 November 2014
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	====Coordinate and Matrix Representation Problems====		====Coordinate and Matrix Representation Problems====

	Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.

Bug: /* Boris */

2014-11-29T23:51:30Z

Boris

← Older revision		Revision as of 19:51, 29 November 2014
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	Recall:		Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is the column vector \begin{pmatrix}c_1\\c_2\\c_3\\.\\c_n\end{pmatrix}</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = \{v_1, v_2, v_3, ..., v_n\}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is the column vector <math> \begin{pmatrix}c_1\\c_2\\c_3\\...\\c_n\end{pmatrix}</math>.


	Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math>, <math>B = {v_1, v_2, v_3, ..., v_m}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then the matrix representation of <math>T</math> in the ordered bases <math>B, K</math> is the matrix		Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math> and <math>K = \{v_1, v_2, v_3, ..., v_m\}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then <math>T(v_j) = \displaystyle\sum_{i=1}^{m} c_{ij}T(v_j)</math> where <math>c_{ij} \in F</math>. Then the '''matrix representation''' of <math>T</math> in the ordered bases <math>B, K</math> is the matrix <math>\begin{pmatrix}c_{11} & c_{12} & c_{13} & ... & c_{1n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\c_{31} & c_{32} & c_{33} & ... & c_{3n}\\c_{21} & c_{22} & c_{23} & ... & c_{2n}\\... & ... & ... & ...& ... \\c_{m1} & c_{m2} & c_{m3} & ... & c_{mn}\end{pmatrix}</math>.

Bug at 23:42, 29 November 2014

2014-11-29T23:42:13Z

← Older revision		Revision as of 19:42, 29 November 2014
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	Recall:		Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>~~. Let~~ <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is ~~defined~~ by ~~<math>[v]_B =~~ </math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>, <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is the column vector \begin{pmatrix}c_1\\c_2\\c_3\\.\\c_n\end{pmatrix}</math>.

			Let <math>W</math> be a finite dimensional vector space over the same field <math>F</math>, <math>B = {v_1, v_2, v_3, ..., v_m}</math> be an ordered basis of <math>W</math>. Define a linear transformation <math>T:V \to W</math>. Then the matrix representation of <math>T</math> in the ordered bases <math>B, K</math> is the matrix

Bug at 23:26, 29 November 2014

2014-11-29T23:26:00Z

← Older revision		Revision as of 19:26, 29 November 2014
Line 7:		Line 7:
	Recall:		Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is defined by <math>[v]_B = ~~[c_1, c_2, c_3, ..., c_n]~~</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is defined by <math>[v]_B = </math>.

Bug: /* Coordinate and Matrix Representation Problems */

2014-11-29T23:24:03Z

Coordinate and Matrix Representation Problems

← Older revision		Revision as of 19:24, 29 November 2014
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	Recall:		Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>V</math> is defined by <math>[v]_B = [c_1, c_2, c_3, ..., c_n]</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>v</math> is defined by <math>[v]_B = [c_1, c_2, c_3, ..., c_n]</math>.

Bug: /* Coordinate and Matrix Representation Problems */

2014-11-29T23:23:47Z

Coordinate and Matrix Representation Problems

← Older revision		Revision as of 19:23, 29 November 2014
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	Recall:		Recall:

	Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> ~~for~~ <math>c_i \in F</math> ~~and~~ the coordinate representation of <math>V</math> is defined by <math>[v]_B = [c_1, c_2, c_3, ..., c_n]</math>.		Let <math>V</math> be a finite dimensional vector space over a field <math>F</math>. Let <math>B = {v_1, v_2, v_3, ..., v_n}</math> be an ordered basis of <math>V</math> and <math>v \in V</math>. Then <math>v = \displaystyle\sum_{i=1}^{n} c_iv_i</math> where <math>c_i \in F</math>. Then the '''coordinate representation''' of <math>V</math> is defined by <math>[v]_B = [c_1, c_2, c_3, ..., c_n]</math>.