Lent Term Week 1 (lectures 21 and 22): In this question, you will look at the null space, column space, and row space of a matrix; and linear independence, spanning sets, and bases.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
This will be used in the definition of the vector b1.
", "definition": "random(-5..5 except(0))", "name": "tt", "group": "Variables for parts f and g"}, "v1": {"templateType": "anything", "description": "This will be one of our vectors in part b.
", "definition": "vector(a , b , c , d , f)", "name": "v1", "group": "Variables for parts a to e"}, "l": {"templateType": "anything", "description": "This is one of the entries of the vector u1.
", "definition": "random(-5..5 except(0))", "name": "l", "group": "Variables for parts f and g"}, "w3": {"templateType": "anything", "description": "This is needed in the definition of the vector v3. It is linearly independent to v1.
", "definition": "vector(0 , g , h , j , k)", "name": "w3", "group": "Variables for parts a to e"}, "RRE_A_ef": {"templateType": "anything", "description": "This is the reduced row echelon form of A_ef
", "definition": "matrix([1 , uu , 0 , 1] , [0 , 0 , 1 , rr] , [0 , 0 , 0 , 0])", "name": "RRE_A_ef", "group": "Variables for parts f and g"}, "t": {"templateType": "anything", "description": "This is needed in the definition of the vector v2.
", "definition": "random(-4..4 except(0) except(1) except(-1))", "name": "t", "group": "Variables for parts a to e"}, "k": {"templateType": "anything", "description": "This will be one of the entries in the vector w2.
", "definition": "random(-5..5 except(0))", "name": "k", "group": "Variables for parts a to e"}, "m": {"templateType": "anything", "description": "This is one of the entries of the vector u1.
", "definition": "random(-5..5 except(0))", "name": "m", "group": "Variables for parts f and g"}, "v4": {"templateType": "anything", "description": "This will be one of our vectors in part b. It is linearly dependent on v1 and v3.
", "definition": "v1 + r*v3", "name": "v4", "group": "Variables for parts a to e"}, "s": {"templateType": "anything", "description": "This is needed in the definition of the vector v3.
", "definition": "random(-4..4 except(0) except(1) except(-1))", "name": "s", "group": "Variables for parts a to e"}, "u2": {"templateType": "anything", "description": "This is one of the vectors in part f. It is a scalar multiple of u1.
", "definition": "uu * u1", "name": "u2", "group": "Variables for parts f and g"}, "RRE_A": {"templateType": "anything", "description": "This is the RRE form of the matrix A whose colums are v1 , v2 , v3 , v4.
", "definition": "matrix([1 , t , 0 , 1] , [0 , 0 , 1 , r], [0 , 0 , 0 , 0] , [0 , 0 , 0 , 0] , [0 , 0 , 0 , 0])", "name": "RRE_A", "group": "Variables for parts a to e"}, "f": {"templateType": "anything", "description": "This will be one of the entries in the vector v1.
", "definition": "random(-5..5 except(0))", "name": "f", "group": "Variables for parts a to e"}, "h": {"templateType": "anything", "description": "This will be one of the entries in the vector w2.
", "definition": "random(-5..5 except(0))", "name": "h", "group": "Variables for parts a to e"}, "A_ef_b1": {"templateType": "anything", "description": "This is the matrix A_ef augmented with the vector b1.
", "definition": "matrix([l , uu*l , 0 , l , ss*l] , [m , uu*m , q , m +rr*q , ss*m + tt*q] , [n , uu*n , p , n +rr*p , ss*n + tt*p])", "name": "A_ef_b1", "group": "Variables for parts f and g"}, "uu": {"templateType": "anything", "description": "This will be used in the definition of the vector u2.
", "definition": "random(-4..4 except(0) except(1) except(-1))", "name": "uu", "group": "Variables for parts f and g"}, "j": {"templateType": "anything", "description": "This will be one of the entries in the vector w2.
", "definition": "random(-5..5 except(0))", "name": "j", "group": "Variables for parts a to e"}, "c": {"templateType": "anything", "description": "This will be one of the entries in the vector v1.
", "definition": "random(-5..5 except(0))", "name": "c", "group": "Variables for parts a to e"}, "RRE_A_ef_b1": {"templateType": "anything", "description": "This is the RRE form of A_ef_b1
", "definition": "matrix([1 , uu , 0 , 1 , ss] , [0 , 0 , 1 , rr , tt] , [0 , 0 , 0 , 0 , 0])", "name": "RRE_A_ef_b1", "group": "Variables for parts f and g"}, "a": {"templateType": "anything", "description": "This will be one of the entries in the vector v1.
", "definition": "random(-5..5 except(0))", "name": "a", "group": "Variables for parts a to e"}, "v2": {"templateType": "anything", "description": "This will be one of our vectors in part b. It is a scalar multiple of v1.
", "definition": "t*v1", "name": "v2", "group": "Variables for parts a to e"}, "rr": {"templateType": "anything", "description": "This will be used in the definition of u4.
", "definition": "random(-4..4 except(0) except(1) except(-1))", "name": "rr", "group": "Variables for parts f and g"}, "u1": {"templateType": "anything", "description": "This is one of the vector that we use in part f.
", "definition": "vector(l , m , n)", "name": "u1", "group": "Variables for parts f and g"}, "ss": {"templateType": "anything", "description": "This will be used in the definition of the vector b1.
", "definition": "random(-5..5 except(0))", "name": "ss", "group": "Variables for parts f and g"}, "g": {"templateType": "anything", "description": "This will be one of the entries in the vector w2.
", "definition": "random(-5..5 except(0))", "name": "g", "group": "Variables for parts a to e"}, "v3": {"templateType": "anything", "description": "This will be one of our vectors in part b. It is linearly independent to v1 and v2.
", "definition": "v1 + s*w3", "name": "v3", "group": "Variables for parts a to e"}, "q": {"templateType": "anything", "description": "This is one of the entries of the vector u2.
", "definition": "random(-5..5 except(0))", "name": "q", "group": "Variables for parts f and g"}, "n": {"templateType": "anything", "description": "This is one of the entries of the vector u1.
", "definition": "random(-5..5 except(0))", "name": "n", "group": "Variables for parts f and g"}, "d": {"templateType": "anything", "description": "This will be one of the entries in the vector v1.
", "definition": "random(-5..5 except(0))", "name": "d", "group": "Variables for parts a to e"}, "b": {"templateType": "anything", "description": "This will be one of the entries in the vector v1.
", "definition": "random(-5..5 except(0))", "name": "b", "group": "Variables for parts a to e"}, "p": {"templateType": "anything", "description": "This is one of the entries of the vector u2.
", "definition": "random(-5..5 except(0))", "name": "p", "group": "Variables for parts f and g"}, "A_ef": {"templateType": "anything", "description": "This is the matrix A from parts e and f (Hence the variable name A_ef).
", "definition": "matrix([l , uu*l , 0 , l] , [m , uu*m , q , m +rr*q] , [n , uu*n , p , n +rr*p])", "name": "A_ef", "group": "Variables for parts f and g"}, "b1": {"templateType": "anything", "description": "This is avector that is used in parts e and f.
", "definition": "ss*u1 + tt*u3", "name": "b1", "group": "Variables for parts f and g"}, "u4": {"templateType": "anything", "description": "This is one of the vectors of part f. It is dependent on u1 and u3.
", "definition": "u1 + rr*u3", "name": "u4", "group": "Variables for parts f and g"}, "u3": {"templateType": "anything", "description": "This is one of the vectors in part f. It is linearly independent to u1 and u2.
", "definition": "vector(0 , q , p)", "name": "u3", "group": "Variables for parts f and g"}, "r": {"templateType": "anything", "description": "This is needed in the definition of the vector v4.
", "definition": "random(-4..4 except(0) except(1) except(-1))", "name": "r", "group": "Variables for parts a to e"}}, "advice": "", "variable_groups": [{"variables": ["a", "b", "c", "d", "f", "v1", "t", "v2", "g", "h", "j", "k", "w3", "s", "v3", "r", "v4", "RRE_A"], "name": "Variables for parts a to e"}, {"variables": ["l", "m", "n", "q", "p", "u1", "uu", "u2", "u3", "rr", "u4", "A_ef", "RRE_A_ef", "ss", "tt", "b1", "A_ef_b1", "RRE_A_ef_b1"], "name": "Variables for parts f and g"}], "preamble": {"css": "", "js": ""}, "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "unitTests": [], "prompt": "Let $A$ be a real $m \\times n$ matrix. By choosing one option for each row, complete the following table to demonstrate the definitions of the vector spaces $N(A)$, $CS(A)$, and $RS(A)$.
\n[[0]]
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", "The linear span of the columns of $A$.
", "The linear span of the transposed rows of $A$.
"], "showFeedbackIcon": true, "shuffleAnswers": true, "scripts": {}, "warningType": "none", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minAnswers": 0, "displayType": "radiogroup", "maxAnswers": 0, "marks": 0, "variableReplacements": [], "shuffleChoices": true, "choices": ["$N(A)$
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$\\boldsymbol{v1} = \\var{{v1}} , \\boldsymbol{v2} = \\var{{v2}} , \\boldsymbol{v3} = \\var{{v3}} , \\boldsymbol{v4} = \\var{{v4}}$.
Find the reduced row echelon form of the matrix $A = ( \\boldsymbol{v1} \\; \\boldsymbol{v2} \\;\\boldsymbol{v3} \\; \\boldsymbol{v4} )$, whose columns are the vectors in $X$.
\n$\\textbf{RRE} (A) =$ [[0]] .
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\nA matrix has full row rank if the row rank is equal to the number of rows. A matrix has full column rank if the column rank is equal to the number of columns. While the row rank and column rank are equal, the number of rows and number of columns may not be. Hence, a matrix may have full row rank but not have full column rank, or vice versa.
\nIf an $m \\times n$ matrix has full row rank then its column vectors span $\\mathbb{R}^m$. If an $m \\times n$ matrix has full column rank then its column vectors are linearly independent.
", "type": "information", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": []}], "variableReplacements": [], "prompt": "For this part of the question, if you need a hint press the \"Shows steps\" button.
\nWe can see that the row rank of $A$ is [[0]] and so $A$ [[1]] have full row rank. Hence, the set $X$ (whose vectors are the columns of $A$) [[2]] span $\\mathbb{R}^5$.
\nWe can see that the column rank of $A$ is [[3]] and so $A$ [[4]] have full column rank. Hence, the set $X$ (whose vectors are the columns of $A$) [[5]] linearly independent.
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\nWe can see that the columns of $A$ with leading $1\\text{s}$ are columns [[0]] and [[1]]. (Please enter an integer such as 1, 2, 3, or 4).
\nHence, considering that the vectors in $X$ are the columns of the matrix $A$, we can obtain, without any calculations, a basis $B$ for $\\text{Lin} (X)$:
\n$B= \\Bigg\\{$ [[2]] $,$ [[3]] $\\Bigg\\}$
\nThe dimension of $\\text{Lin} (X)$ is [[4]].
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\nYou may do this by expressing these equations in matrix form and then solving them by finding the reduced row echelon form of the augmented matrix, or you may simply use insepction.
\n$\\lambda_1 =$ [[0]] .
$\\lambda_1 =$ [[1]] .
$\\mu_1 =$ [[2]] .
$\\mu_2 =$ [[3]] .
Now consider the linear system $A \\boldsymbol{x} = \\boldsymbol{b}$, where
$A = \\var{{A_ef}}$ , $\\boldsymbol{x} = \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ x_4 \\end{pmatrix}$, and $\\boldsymbol{b} = \\var{{b1}}$.
Find the reduced row echelon form of $(A | \\boldsymbol{b})$.
$\\textbf{RRE} (A | \\boldsymbol{b}) =$ [[0]] .
From $\\textbf{RRE} (A | \\boldsymbol{b})$ we can see that the system $A \\boldsymbol{x} = \\boldsymbol{b}$ is consistent, and $x_2$ and $x_4$ are free variables. Let $x_2 = s$ and $x_4 = t$. Then, $\\textbf{RRE} (A | \\boldsymbol{b})$ tells us that the general solution of $A \\boldsymbol{x} = \\boldsymbol{b}$ is
$\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ x_4 \\end{pmatrix} =$ [[1]] $+$ [[2]] $s +$ [[3]] $t$.
Let $\\boldsymbol{c_i}$ represent the $i^{\\text{th}}$ column of $A$.
\nWe can see that the first and third columns of $\\textbf{RRE} (A | \\boldsymbol{b})$ have leading $1\\text{s}$. Hence, $\\{ \\boldsymbol{c_1} , \\boldsymbol{c_3} \\}$ is a basis for the column space of $A$.
\nExpress $\\boldsymbol{c_2} , \\boldsymbol{c_4},$ and $\\boldsymbol{b}$ in terms of $\\boldsymbol{c_1}$ and $\\boldsymbol{c_3}$. (This is similar to part e above)
\n$\\boldsymbol{c_2} =$ [[0]] $\\boldsymbol{c_1} +$ [[1]] $\\boldsymbol{c_3}$.
$\\boldsymbol{c_4} =$ [[2]] $\\boldsymbol{c_1} +$ [[3]] $\\boldsymbol{c_3}$.
$\\boldsymbol{b} =$ [[4]] $\\boldsymbol{c_1} +$ [[5]] $\\boldsymbol{c_3}$.
This is the question for Lent Term week 1 of the MA100 course at the LSE. It looks at material from chapters 21 and 22.
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