Links to other worksheets:
Book, (1) Linear Algebra Arithmetic, (2) Linear Algebra Applications, (4) Linear Transformations, (5) Changing Bases
You can evaluate cells by typing Shift+Enter (or clicking in them and clicking "evaluate"). Do this on the 2+3 cell below.
You can add new input cells in which to type Sage commands by clicking above or below each cell when a horizontal bar appears. Try adding a new cell above and below the 2+3 cell.
You can add new text cells by holding down shift when you click to enter a new cell. Try doing this on the cell below.
You can edit existing text cells by double clicking them. Try modifying this text cell by double clicking on it.
︡f074ea8a-77e8-4afa-a42d-7d8a6b9b58bb︡{"html": "Links to other worksheets:
\nBook, (1) Linear Algebra Arithmetic, (2) Linear Algebra Applications, (4) Linear Transformations, (5) Changing Bases
\nYou can evaluate cells by typing Shift+Enter (or clicking in them and clicking \"evaluate\"). Do this on the 2+3 cell below.
\nYou can add new input cells in which to type Sage commands by clicking above or below each cell when a horizontal bar appears. Try adding a new cell above and below the 2+3 cell.
\nYou can add new text cells by holding down shift when you click to enter a new cell. Try doing this on the cell below.
\nYou can edit existing text cells by double clicking them. Try modifying this text cell by double clicking on it.
"}︡ ︠60168b6c-b230-47c8-b1a5-b3245bfce0cd︠ 2+3 ︡3a9870ab-a4ab-403c-a379-f57817214e06︡{"stdout": "5"}︡ ︠6619f946-9513-4fd8-86a8-617cf8875a25i︠ %htmlNow you're ready to begin.
Use the vector command to enter vectors. Functions always require paranthesis ( ) and lists always require brackets [ ].
︡1e687f28-71b0-4764-b1e2-c1f401e7eba2︡{"html": "Now you're ready to begin.
\nUse the vector command to enter vectors. Functions always require paranthesis ( ) and lists always require brackets [ ].
"}︡ ︠2ae9b7d1-6372-4584-b74e-de823de0ba83︠ u=vector([1,2]) u ︡818226ef-5fe2-4cf7-b330-f476700ff27c︡{"stdout": "(1, 2)"}︡ ︠dbe654f4-d69d-451c-af54-96446775d41b︠ v=vector([3,1]) v ︡16b880ad-c965-4baf-985e-a29d81c4a22e︡{"stdout": "(3, 1)"}︡ ︠fdc996d2-a0f9-4f97-b95a-f0ce6f474c10i︠ %htmlThe magnitude of a vector is also called the length or norm. Use .norm() to find lengths in Sage.
︡6d05d074-b793-4dd1-8570-91762baf79df︡{"html": "The magnitude of a vector is also called the length or norm. Use .norm() to find lengths in Sage.
"}︡ ︠efb2ce30-9f08-4752-9b27-6186cb92634f︠ u.norm() ︡f7b1fdc2-9d3b-4450-aa57-f21531cec8e5︡{"stdout": "sqrt(5)"}︡ ︠21229e71-2dcc-4ecb-88ca-c1c29b6f554fi︠ %htmlThe dot product is simply an asterisk *.
︡85a29f2c-0a61-4b8b-af6a-f6a98dd26932︡{"html": "The dot product is simply an asterisk *.
"}︡ ︠ecc06a42-ddcf-494a-8bbb-f5c2caf5925a︠ u*v ︡667f92a8-ab9a-4cfa-9f80-d363159e55a1︡{"stdout": "5"}︡ ︠2e822934-cf34-4870-a62b-d85bb5a06136i︠ %htmlHere is an example of how to plot vectors. This works in both 2D and 3D.
︡41818db1-4f06-4028-8f44-d7fdd8367363︡{"html": "Here is an example of how to plot vectors. This works in both 2D and 3D.
"}︡ ︠599765df-0bc5-472d-abc8-58aee1267307︠ plot(u)+plot(v) ︡afdef4ab-62c5-4e1d-ab88-a408b010acfd︡{"html": "
"}︡
︠32217611-1ab4-467a-aa79-d84d3e4b14e0i︠
%html
To create an arrow with its base at $v$ and head at $u$, use arrow(v,u). To get this to work in 3D, you have to use the command arrow3d(v,u) instead of just arrow(v,u).
︡bbc729c8-7123-453a-9f32-6def7223d4ee︡{"html": "To create an arrow with its base at $v$ and head at $u$, use arrow(v,u). To get this to work in 3D, you have to use the command arrow3d(v,u) instead of just arrow(v,u).
"}︡ ︠e405b67e-98f2-4c6b-86ad-783327530ba0︠ plot(u)+plot(v)+arrow(u,u+v)+plot(u+v,color='black') ︡4810a028-e4ab-4c3d-9f59-9f8c28dda15b︡{"html": ""}︡
︠ca4993ab-d5ca-4219-a4ee-4ae245801a3b︠
plot(u)+plot(v)+plot(u-v,color='black')+arrow(v,u,color='black',linestyle='dashed')
︡6b68d26b-3337-4a09-89a6-dd5f58c05bd8︡{"html": ""}︡
︠b6aa063a-c2d5-48d8-bcb3-d2531f8ad973i︠
%html
Here's the 3d code, put in a nice html table. If you get an error below, just reevaluate this line of code.
︡4f258bc9-e126-494d-919a-fd103ddc0188︡{"html": "Here's the 3d code, put in a nice html table. If you get an error below, just reevaluate this line of code.
"}︡ ︠6299cf07-c005-4f2b-8fc4-1d73c4c23e38︠ u=vector([1,2,3]) v=vector([3,1,0]) html.table([ ["u", u], ["v", v], ["|u|", u.norm()], ["|v|", v.norm()], ["u.v", u*v], ]) plot(u)+plot(v) ︡cfda7fa4-9435-4ae1-8851-9bfcd98cd55a︡{"html": "\n| u | \n\\left(1,2,3\\right) | \n
| v | \n\\left(3,1,0\\right) | \n
| |u| | \n\\sqrt{14} | \n
| |v| | \n\\sqrt{10} | \n
| u.v | \n5 | \n
To enter a matrix, use the matrix command. To store the matrix using a name, use A = matrix( ... ). Here is an example.
︡36cbc303-157f-438f-98fd-0745d7f5f731︡{"html": "To enter a matrix, use the matrix command. To store the matrix using a name, use A = matrix( ... ). Here is an example.
"}︡ ︠7fb0630d-c6df-424e-aa64-7c3da2a4150c︠ A = matrix([[-1,2,3,4],[4,5,6,0],[3,2,8,-4]]) A ︡14814759-d531-402f-ba50-a0081c0050ef︡{"stdout": "[-1 2 3 4]\n[ 4 5 6 0]\n[ 3 2 8 -4]"}︡ ︠6e890aca-7ceb-4354-867a-7f70d7a66a89i︠ %htmlTo row reduce the matrix, type the name of the matrix, a period, and then type echelon_form(). If you are familar with tab completion, Sage includes tab completion. This means that you can type "A.ec" and then type "Tab" and it will show you all the commands that start with "ec" which can be applied to the matrix A. The command .rref() is availalbe in Sage version 4.3.4, so the next official release will allow us to type A.rref().
︡e8213185-c33c-45be-a350-55a05bbc4abc︡{"html": "To row reduce the matrix, type the name of the matrix, a period, and then type echelon_form(). If you are familar with tab completion, Sage includes tab completion. This means that you can type \"A.ec\" and then type \"Tab\" and it will show you all the commands that start with \"ec\" which can be applied to the matrix A. The command .rref() is availalbe in Sage version 4.3.4, so the next official release will allow us to type A.rref().
"}︡ ︠e7d90c1f-c2d7-429b-bbbf-2493b5b7a78b︠ A.echelon_form() ︡2af6ff5e-11cd-44ab-a99b-6e836b6d90a0︡{"stdout": "[ 1 0 59 -20]\n[ 0 1 31 -8]\n[ 0 0 77 -24]"}︡ ︠7cff6758-ec21-4a56-b563-1997f7120effi︠ %htmlNotice that the 77 above is not a leading 1. This is because Sage keeps track of the number system behind the matrix. If you type in a matrix with all integer entries, then Sage will assume that all operations are to be done using only integers. It will not fully reduce a matrix if the matrix requires division that will result in fractions. To fix this, you can tell Sage that you want to work with fractions by typing QQ at the beginning of your matrix. QQ is the Sage symbol for "Rational Numbers" or $\mathbb Q$
︡2c959c26-91d9-48b5-959b-ba852f941b75︡{"html": "Notice that the 77 above is not a leading 1. This is because Sage keeps track of the number system behind the matrix. If you type in a matrix with all integer entries, then Sage will assume that all operations are to be done using only integers. It will not fully reduce a matrix if the matrix requires division that will result in fractions. To fix this, you can tell Sage that you want to work with fractions by typing QQ at the beginning of your matrix. QQ is the Sage symbol for \"Rational Numbers\" or $\\mathbb Q$
"}︡ ︠2c0ceb8b-a430-4afc-993f-8099e2cd801f︠ A = matrix(QQ,[[-1,2,3,4],[4,5,6,0],[3,2,8,-4]]) A ︡4101c722-5433-43cf-96e4-7c8d8190e667︡{"stdout": "[-1 2 3 4]\n[ 4 5 6 0]\n[ 3 2 8 -4]"}︡ ︠f45ba0e1-2257-4014-b3b0-c9aef02659a1︠ A.echelon_form() ︡ddf209ab-dc5e-4854-ac65-9f13786b1236︡{"stdout": "[ 1 0 0 -124/77]\n[ 0 1 0 128/77]\n[ 0 0 1 -24/77]"}︡ ︠bc7ff38c-144c-4502-b352-82800a27e731i︠ %htmlI prefer the following way to enter matrices. Type QQ to allow fractional arithmetic, then the number of rows, numbers of columns, and then a list of the entries read left to right, top to bottom. You can eliminate writing lots of brackets this way.
︡dadd70ef-bc7f-42e0-a4b5-12834703453f︡{"html": "I prefer the following way to enter matrices. Type QQ to allow fractional arithmetic, then the number of rows, numbers of columns, and then a list of the entries read left to right, top to bottom. You can eliminate writing lots of brackets this way.
"}︡ ︠c4c98811-2fba-4aab-b71c-1fd6e2709326︠ A=matrix(QQ,2,3,[1,2,3,4,5,6]) A ︡8885fb1d-af92-4e7f-bdc7-c7b20ad2e775︡{"stdout": "[1 2 3]\n[4 5 6]"}︡ ︠bee15349-24bd-4845-ae23-78c291300157i︠ %htmlIf you want to make a matrix of vectors, Sage will quickly place the vectors into rows. To get them to be columns, use the transpose command.
︡ed7aa8c5-edff-4fa8-baf6-a3311d7c5e3d︡{"html": "If you want to make a matrix of vectors, Sage will quickly place the vectors into rows. To get them to be columns, use the transpose command.
"}︡ ︠d19de006-f3b7-406b-82de-0cbda2cb812c︠ u=vector([1,2]) v=vector([3,4]) w=vector([5,6]) A=matrix(QQ,[u,v,w]) A ︡e3146ba8-f67e-4c04-b642-94541ddeb942︡{"stdout": "[1 2]\n[3 4]\n[5 6]"}︡ ︠fd2e1957-590c-4a26-b7d6-0b43b07a6670︠ B=A.transpose() B ︡a2d1b4e4-2cef-4915-bb60-131f36e9fc48︡{"stdout": "[1 3 5]\n[2 4 6]"}︡ ︠48a9f788-93b7-4223-8f70-b6d5511c3f97i︠ %htmlYou can access specific entries in a matrix by typing A[i,j]. However, note that Sage begins all numbering at 0 instead of 1. So the $a_{11}$ spot in the matrix is $A[0,0]$.
︡bd145039-3052-47ed-a39a-5ad698fbc7c8︡{"html": "You can access specific entries in a matrix by typing A[i,j]. However, note that Sage begins all numbering at 0 instead of 1. So the $a_{11}$ spot in the matrix is $A[0,0]$.
"}︡ ︠f5b4e06a-41b8-4c82-a153-90a12551c99f︠ A=matrix(QQ,2,3,[1,2,3,4,5,6]) A ︡49e4bf36-86e2-43a6-8dcf-5e08c76027ce︡{"stdout": "[1 2 3]\n[4 5 6]"}︡ ︠0e5da498-3ffd-41a3-857c-41ffef7432ce︠ A[0,0] ︡d03bf76a-50e4-4fc1-abef-4fedc9f61c08︡{"stdout": "1"}︡ ︠ccd48e3d-71d1-450f-8d9e-53d893593a91︠ A[0] ︡698b5dd4-e266-4ba4-8db6-505756651ef8︡{"stdout": "(1, 2, 3)"}︡ ︠bbb1aff5-c979-4f21-bb9c-49448b378aa2︠ A[1,2] ︡581b07af-624c-4cf5-a0fb-ba1616b6f785︡{"stdout": "6"}︡ ︠d8c5f79b-5540-487f-a038-b6420c7b9f22i︠ %htmlJust enter both matrices, and then use an asterisk * to multiply them.
︡09133516-a34c-463d-97ed-dae3528d3056︡{"html": "Just enter both matrices, and then use an asterisk * to multiply them.
"}︡ ︠8d456070-8247-4829-9225-6fb4da51af95︠ A=matrix(QQ,2,3,[1,2,3,4,5,6,]) A ︡2a5e0a51-5599-4c40-adff-84eeee9cd931︡{"stdout": "[1 2 3]\n[4 5 6]"}︡ ︠f454800e-47d1-4361-a259-3171fc400f26︠ B=matrix(QQ,3,2,[1,2,3,4,5,6,]) B ︡6088d3cc-b5be-4f96-bfc9-5e78a54d5220︡{"stdout": "[1 2]\n[3 4]\n[5 6]"}︡ ︠3d571230-1e1a-4fb6-9ec4-e2d4ffda0395︠ A*B ︡38b39dab-8cd2-4286-a228-0478faa6c7ea︡{"stdout": "[22 28]\n[49 64]"}︡ ︠695c3092-f9cd-47b5-a4e5-8b35358cbeac︠ B*A ︡6106945e-f8ad-47dc-b1cc-d88f71f9e8b8︡{"stdout": "[ 9 12 15]\n[19 26 33]\n[29 40 51]"}︡ ︠02aa8b25-bc8e-42a3-bc51-d5e70ca1b696i︠ %htmlYou'll get an error if matrix multiplication doesn't make sense.
︡45b6e293-7b6d-4a47-a3b0-bd6c38afd3b6︡{"html": "You'll get an error if matrix multiplication doesn't make sense.
"}︡ ︠e9e9f118-08a0-4e7a-96af-5f5e1c479154︠ A*A ︡062a3c48-208e-46f9-b3e3-6bb26b394d6e︡{"stderr": "Traceback (most recent call last):\n File \"Most commands we will need all semester long are available using tab completion. First enter your matrix, then type A and a period, and then type the "Tab" key. This will bring up a list of commands. Try doing this now in the blank cell below (after evaluating the first cell by clicking in it and evaluating it (type Shift+Enter).
︡08a3698f-93d5-4f44-8df3-6133c250cdff︡{"html": "Most commands we will need all semester long are available using tab completion. First enter your matrix, then type A and a period, and then type the \"Tab\" key. This will bring up a list of commands. Try doing this now in the blank cell below (after evaluating the first cell by clicking in it and evaluating it (type Shift+Enter).
"}︡ ︠9b8d9a01-5cdb-4165-b87a-8d61fdf0e956︠ A=matrix(QQ,3,3,[2,1,0,1,2,0,0,0,1]) A ︡40406ea8-7d33-4df6-b705-12a5d1d3b903︡{"stdout": "[2 1 0]\n[1 2 0]\n[0 0 1]"}︡ ︠ce066e70-77ab-475c-82e7-0e0da20bd95ai︠ %htmlWe'll be using the rank(), det(), inverse(), eigenvalues(), and eigenvectors_right() commands.
︡f6d6d7cc-611c-4af8-85f7-790cda73d528︡{"html": "We'll be using the rank(), det(), inverse(), eigenvalues(), and eigenvectors_right() commands.
"}︡ ︠c14a846a-aaf2-41f7-8ff1-2f5c753424c2︠ A.rank() ︡8509e8dc-2563-4c43-99a6-1a889423e4af︡{"stdout": "3"}︡ ︠317a3581-824f-4b43-aa1e-94ae727dd731︠ A.det() ︡92ec81f1-d522-4b73-9448-a76b6fac0f6b︡{"stdout": "3"}︡ ︠9bdcae94-05ec-426b-9a98-aab20c0f89ea︠ A.echelon_form() ︡ffe896b6-f275-4e49-90c3-9229517b150e︡{"stdout": "[1 0 0]\n[0 1 0]\n[0 0 1]"}︡ ︠b89a46e0-438e-4528-9fc6-8e74b7383fbb︠ A.inverse() ︡c3afb824-1fb1-4dcc-b899-da1dbdabfdb6︡{"stdout": "[ 2/3 -1/3 0]\n[-1/3 2/3 0]\n[ 0 0 1/3]"}︡ ︠d8489545-be59-434e-96b9-5e7e6ff57cc6︠ A.eigenvalues() ︡07694994-b4d6-4e99-8229-ef4876df1007︡{"stdout": "[3, 1, 1]"}︡ ︠429db4ea-fe86-4c6d-b8ec-f02c903976ed︠ A.eigenvectors_right() ︡f17fbf36-b96a-487b-bd3e-0ed467b3dc92︡{"stdout": "[(3, [\n(1, 1, 0)\n], 1), (1, [\n(1, -1, 0),\n(0, 0, 1)\n], 2)]"}︡ ︠aa522b13-fbf9-4123-9c1d-ab904ce36e00i︠ %htmlTwo questions:
We use "right" because we are trying to solve $A\vec x = \lambda \vec x$ instead of $\vec x A = \vec x \lambda$. The latter are called "left" eigenvectors.
To read the output, you are given a list of values for each eigenvalue. In particular, you are given (the eigenvalue, a set of linearly independent eigenvectors whose span gives all the eigenvalues, and the multiplicity of the eigenvalue - how many times it is a root).
[(3, [ -the eigenvalue︡70ec249d-0aee-4fc8-8912-9fc68d85c55f︡{"html": "
(1, 1, 0) -an eigenvector whose span gives all the rest
], 1), -3 has multiplicity 1.
(1, [ -1 is the next eigenvalue
(1, -1, 0), -Here there are 2 independent eigenvectors whose span gives the rest
(0, 0, 1)
], 2)] -$\lambda=1$ has multiplicty 2
Two questions:
\nWe use \"right\" because we are trying to solve $A\\vec x = \\lambda \\vec x$ instead of $\\vec x A = \\vec x \\lambda$. The latter are called \"left\" eigenvectors.
\nTo read the output, you are given a list of values for each eigenvalue. In particular, you are given (the eigenvalue, a set of linearly independent eigenvectors whose span gives all the eigenvalues, and the multiplicity of the eigenvalue - how many times it is a root).
\n[(3, [ -the eigenvalue"}︡ ︠d1fa1758-7740-4d3d-bbbe-4faa02eea304i︠ %html
(1, 1, 0) -an eigenvector whose span gives all the rest
], 1), -3 has multiplicity 1.
(1, [ -1 is the next eigenvalue
(1, -1, 0), -Here there are 2 independent eigenvectors whose span gives the rest
(0, 0, 1)
], 2)] -$\\lambda=1$ has multiplicty 2
This final example just computes all the values above for a square matrix, and displays them in an easy to read format. If the eigenvalues are rather complicated, then the display may get ugly and you are better off using the commands above.
︡5bbb54f5-6f07-4236-8a88-48881fc41b4d︡{"html": "This final example just computes all the values above for a square matrix, and displays them in an easy to read format. If the eigenvalues are rather complicated, then the display may get ugly and you are better off using the commands above.
"}︡ ︠4c85a11e-cd60-40a1-a7a4-4b7dac0603f1︠ A=matrix(QQ,3,3,[2,1,0,1,2,0,0,4,3]) EV=A.eigenvectors_right() if A.det()==0: inv="No Inverse Exists" ; else: inv=A.inverse(); list=([ ["Matrix",A], ["Transpose",A.transpose()], ["RREF",A.echelon_form()], ["Rank",A.rank()], ["Determinant",A.det()], ["Inverse",inv], ["Eigenvalues",A.eigenvalues()],[],[],[],[], ["Eigenvalue","Linearly Independent Eigenvectors", "Multiplicity of Eigenvalue"]]) for n in range(len(EV)): list.append(EV[n]) html.table(list) ︡bf01264c-33fe-4a4d-8e62-467e29492196︡{"html": "\n| Matrix | \n\\left(\\begin{array}{rrr}\n2 & 1 & 0 \\\\\n1 & 2 & 0 \\\\\n0 & 4 & 3\n\\end{array}\\right) | \n|
| Transpose | \n\\left(\\begin{array}{rrr}\n2 & 1 & 0 \\\\\n1 & 2 & 4 \\\\\n0 & 0 & 3\n\\end{array}\\right) | \n|
| RREF | \n\\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right) | \n|
| Rank | \n3 | \n|
| Determinant | \n9 | \n|
| Inverse | \n\\left(\\begin{array}{rrr}\n\\frac{2}{3} & -\\frac{1}{3} & 0 \\\\\n-\\frac{1}{3} & \\frac{2}{3} & 0 \\\\\n\\frac{4}{9} & -\\frac{8}{9} & \\frac{1}{3}\n\\end{array}\\right) | \n|
| Eigenvalues | \n\\left[1, 3, 3\\right] | \n|
| Eigenvalue | \nLinearly Independent Eigenvectors | \nMultiplicity of Eigenvalue | \n
| 1 | \n\\left[\\left(1,-1,2\\right)\\right] | \n1 | \n
| 3 | \n\\left[\\left(0,0,1\\right)\\right] | \n2 | \n