Suppose we have a finite-dimensional vector space $V$ and let $\\boldsymbol{u} \\in V$. Suppose further that $V$ has a basis $M$. We use the notation $(\\boldsymbol{u})_M$ to denote the coordinates of $\\boldsymbol{u}$ in terms of the basis $M$. This may look something like $(\\boldsymbol{u})_M = \\begin{pmatrix} 5 \\\\ 1 \\end{pmatrix}_M$. However, due to the current limitations of this system, we may occasionally need to write $(\\boldsymbol{u})_M = \\begin{pmatrix} 5 \\\\ 1 \\end{pmatrix} {}_M$ instead.
\nConsider the standard basis $B = \\{ \\boldsymbol{e_1} , \\boldsymbol{e_2} \\}$ where $ \\boldsymbol{e_1} = \\var{{e1}} , \\boldsymbol{e_2} = \\var{{e2}}$, and the basis $C = \\{ \\boldsymbol{f_1} , \\boldsymbol{f_2} \\}$ where $\\boldsymbol{f_1} = \\var{{f1}} ,\\boldsymbol{f_2} = \\var{{f2}}$
\nIn parts a to c we will represent each basis vector in terms of its own basis, and in terms of the other basis
\ni) Let us first express the vectors in $B$ in terms of the vectors in $B$. We can see that
$\\boldsymbol{e_1} =$ [[0]] $\\boldsymbol{e_1} +$ [[1]] $\\boldsymbol{e_2}$,
and so
$(\\boldsymbol{e_1})_B =$ [[2]] ${}_B$
Similarly
$\\boldsymbol{e_2} =$ [[3]] $\\boldsymbol{e_1} +$ [[4]] $\\boldsymbol{e_2}$,
and so
$(\\boldsymbol{e_2})_B =$ [[5]] ${}_B$.
ii) Now let us express the vectors in $C$ in terms of the vectors in $B$. We can see that
$\\boldsymbol{f_1} =$ [[6]] $\\boldsymbol{e_1} +$ [[7]] $\\boldsymbol{e_2}$,
and so
$(\\boldsymbol{f_1})_B =$ [[8]] ${}_B$.
Similarly
$\\boldsymbol{f_2} =$ [[9]] $\\boldsymbol{e_1} +$ [[10]] $\\boldsymbol{e_2}$,
and so
$(\\boldsymbol{f_2})_B =$ [[11]] ${}_B$.
iii) Now let us express the vectors in $C$ in terms of the vectors in $C$. We can see that
$\\boldsymbol{f_1} =$ [[12]] $\\boldsymbol{f_1} +$ [[13]] $\\boldsymbol{f_2}$,
and so
$(\\boldsymbol{f_1})_B =$ [[14]] ${}_C$.
Similarly
$\\boldsymbol{f_2} =$ [[15]] $\\boldsymbol{f_1} +$ [[16]] $\\boldsymbol{f_2}$,
and so
$(\\boldsymbol{f_2})_B =$[[17]] ${}_C$.
Note that we have yet to express the vectors in $B$ in terms of the basis $C$. This is a bit more complicated, and we will need to develop some techniques first.
\nFirst, what is the transition matrix $\\textbf{P}_C$?
(Press the \"show steps\" button if you require a hint).
$\\textbf{P}_C =$ [[0]] .
Now that we have the transition matrix $\\textbf{P}_C$, we can take vectors expressed in terms of the basis $C$, and then express them in terms of the basis $B$. Indeed, for any $\\boldsymbol{v} \\in V$ we have that $\\textbf{P}_C \\; (\\boldsymbol{v})_C = (\\boldsymbol{v})_B$.
\nHowever, we would like to have it the other way around: We want to take vectors expressed in terms of the basis $B$, and then express them in terms of the basis $C$. Afterall, in this part of the question we set out to express the vectors in $B$ in terms of the basis $C$.
\nNotice that we can we can multiply both sides of the equation $\\textbf{P}_C \\; (\\boldsymbol{v})_C = (\\boldsymbol{v})_B$ by $\\textbf{P}_C^{-1}$ to get $(\\boldsymbol{v})_C = \\textbf{P}_C^{-1} \\; (\\boldsymbol{v})_B$. This gives us a way to take vectors expressed in terms of the basis $B$, and then express them in terms of the basis $C$.
\nWhat is $\\textbf{P}_C^{-1}$?
$\\textbf{P}_C^{-1} =$ [[1]]
The transition matrix $\\textbf{P}_C$ must satisfy the following: $\\textbf{P}_C \\; (\\boldsymbol{v})_C = (\\boldsymbol{v})_B$. That is, it must take vectors expressed in terms of the basis $C$, and then express them in terms of the standard basis $B$.
\nIn particular, it must satisfy $\\textbf{P}_C \\; (\\boldsymbol{f_1})_C = (\\boldsymbol{f_1})_B$ and $\\textbf{P}_C \\; (\\boldsymbol{f_2})_C = (\\boldsymbol{f_2})_B$. From part a we know that $(\\boldsymbol{f_1})_C = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}_C$ , $(\\boldsymbol{f_2})_C = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}_C$ , $(\\boldsymbol{f_1})_B = \\begin{pmatrix} \\var{{a}} \\\\ \\var{{b}} \\end{pmatrix}_B$ , $(\\boldsymbol{f_2})_B = \\begin{pmatrix} \\var{{c}} \\\\ \\var{{d}} \\end{pmatrix}_B$, and hence $\\textbf{P}_C$ must satisfy $\\textbf{P}_C \\; \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} \\var{{a}} \\\\ \\var{{b}} \\end{pmatrix}$ and $\\textbf{P}_C \\; \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} =\\begin{pmatrix} \\var{{c}} \\\\ \\var{{d}} \\end{pmatrix}$.
By letting $\\textbf{P}_C = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ and expanding the two equations above, we can easily determine $a,b,c,d$.
Indeed, we obtain $\\textbf{P}_C = \\big((\\boldsymbol{f_1})_B , (\\boldsymbol{f_2})_B \\big)$.
We can now express the vectors, in the basis $B$, in terms of the basis $C$:
\n$(\\boldsymbol{e_1})_C = \\textbf{P}_C^{-1} \\; (\\boldsymbol{e_1})_B = \\textbf{P}_C^{-1} \\; \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} =$ [[0]]${}_C$.
\n$(\\boldsymbol{e_2})_C = \\textbf{P}_C^{-1} \\; (\\boldsymbol{e_2})_B = \\textbf{P}_C^{-1} \\; \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} =$ [[1]]${}_C$.
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": true, "numColumns": 1, "markPerCell": false, "marks": 1, "allowResize": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "numRows": "2", "correctAnswer": "e1_C", "tolerance": 0, "type": "matrix", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": true, "numColumns": 1, "markPerCell": false, "marks": 1, "allowResize": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "numRows": "2", "correctAnswer": "e2_C", "tolerance": 0, "type": "matrix", "customMarkingAlgorithm": "", "unitTests": []}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Finally, let us recap on what we are able to do. We can now take a vector in terms of one of the bases $B$ or $C$, and express it in terms of the other basis:
\nSuppose we are given a vector $\\boldsymbol{w} \\in V$ and we are told that its representation in terms of the basis $C$ is $(\\boldsymbol{w})_C = \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix}_C$. Then its representation in terms of the basis $B$ is
$(\\boldsymbol{w})_B = \\textbf{P}_C \\; (\\boldsymbol{w})_C = \\textbf{P}_C \\; \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix} =$ [[0]]${}_B$.
Now suppose we are given a vector $\\boldsymbol{v} \\in V$ and we are told that its representation in terms of the basis $B$ is $(\\boldsymbol{v})_B = \\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix}_B$. Then its representation in terms of the basis $C$ is
$(\\boldsymbol{v})_C = \\textbf{P}_C^{-1} \\;(\\boldsymbol{v})_B = \\textbf{P}_C^{-1} \\; \\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix} =$ [[1]]${}_C$.
Suppose $A$ is a square matrix. What are the definitions of an eigenvector of $A$ and the corresponding eignenvalue?
\n[[0]]
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""], "choices": ["An eigenvector of the square matrix $A$ is a non-zero vector $\\boldsymbol{v}$ that is stretched by $A$ by a factor of $\\lambda$; i.e. $A \\boldsymbol{v} = \\lambda \\boldsymbol{v}$, $\\boldsymbol{v} \\neq \\boldsymbol{0}$. The strecth factor $\\lambda$ is the corresponding eigenvalue.
", "An eigenvector of the square matrix $A$ is a vector $\\boldsymbol{v}$ that is stretched by $A$ by a factor of $\\lambda \\neq 0$; i.e. $A \\boldsymbol{v} = \\lambda \\boldsymbol{v}$, $\\lambda \\neq 0$. The strecth factor $\\lambda$ is the corresponding eigenvalue.
", "An eigenvector of the square matrix $A$ is a vector $\\boldsymbol{v}$ that is stretched by $A$ by a factor of $\\lambda$; i.e. $A \\boldsymbol{v} = \\lambda \\boldsymbol{v}$. The strecth factor $\\lambda$ is the corresponding eigenvalue.
", "An eigenvector of the square matrix $A$ is a non-zero vector $\\boldsymbol{v}$ that is stretched by $A$ by a factor of $\\lambda \\neq 0$; i.e. $A \\boldsymbol{v} = \\lambda \\boldsymbol{v}$, $\\boldsymbol{v} \\neq \\boldsymbol{0}$, $\\lambda \\neq 0$. The strecth factor $\\lambda$ is the corresponding eigenvalue.
"], "displayType": "radiogroup", "displayColumns": "1", "showFeedbackIcon": true, "showCorrectAnswer": false, "minMarks": 0, "scripts": {}, "type": "1_n_2", "shuffleChoices": true, "marks": 0, "maxMarks": 0, "customMarkingAlgorithm": "", "unitTests": []}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Let $T: \\mathbb{R}^2 \\longrightarrow \\mathbb{R}^2$ be a linear transformation, and consider the standard basis $B = \\{ \\boldsymbol{e_1} , \\boldsymbol{e_2} \\}$ for $\\mathbb{R}^2$ and a non-standard basis $C = \\{ \\boldsymbol{f_1} , \\boldsymbol{f_2} \\}$ for $\\mathbb{R}^2$.
\nSuppose we are told that $T( \\boldsymbol{f_1} ) = \\var{{f}} \\boldsymbol{f_1}$ and $T( \\boldsymbol{f_2} ) = \\var{{g}} \\boldsymbol{f_2}$. Find the matrix $A_T^{C \\rightarrow C}$ representing the transformation $T$ with respect to the basis $C$. (If you require a hint then press the \"Show steps\" button below).
\n$A_T^{C \\rightarrow C} =$ [[0]] .
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "stepsPenalty": 0, "steps": [{"showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "unitTests": [], "variableReplacementStrategy": "originalfirst", "prompt": "Recall from section 26.1 of the lecture notes that $A_T^{C \\rightarrow C} = \\Big( \\big( T( \\boldsymbol{f_1} ) \\big)_C \\; , \\; \\big( T(\\boldsymbol{f_1} ) \\big)_C \\Big)$.
\n$\\big( T(\\boldsymbol{f_1} ) \\big)_C$ is simply the vector representation of $T(\\boldsymbol{f_1} )$ with respect to the basis $C$. For example, suppose $\\boldsymbol{v} \\in V$ and $\\boldsymbol{v} = a \\boldsymbol{f_1} + b \\boldsymbol{f_2}$ for some $a,b \\in \\mathbb{R}$, then $(\\boldsymbol{v})_C = \\begin{pmatrix} a \\\\ b \\end{pmatrix}_C$.
", "marks": 0, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": false, "numColumns": "2", "markPerCell": false, "marks": "3", "allowResize": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "numRows": "2", "correctAnswer": "A_T_CC", "tolerance": 0, "type": "matrix", "customMarkingAlgorithm": "", "unitTests": []}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Suppose further that $\\boldsymbol{f_1} = (\\var{{h}}) \\boldsymbol{e_1} + (\\var{{j}}) \\boldsymbol{e_2}$ and $\\boldsymbol{f_2} = (\\var{{k}}) \\boldsymbol{e_1} + (\\var{{l}}) \\boldsymbol{e_2}$. Find the matrix $A_T^{B \\rightarrow B}$ representing the transformation $T$ with respect respect to the standard basis $B$. (We should note that $A_T^{B \\rightarrow B}$ is occasionally written as $A_T$; this is only done because we are dealing with the standard basis). (Press the \"Show steps\" button if you would like a breakdown of the question into steps).
\n$A_T^{B \\rightarrow B} =$ [[0]] .
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "stepsPenalty": 0, "steps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": false, "prompt": "We recall from section 26.1 of the lecture notes that $A_T^{C \\rightarrow C} = P_C^{-1} \\; A_T^{B \\rightarrow B} \\; P_C$, which we can rearrange as $A_T^{B \\rightarrow B} = P_C \\; A_T^{C \\rightarrow C} \\; P_C^{-1}$.
\nWe have already found $A_T^{C \\rightarrow C}$ in part f. The matrix $P_C$ simply takes vectors in terms of the basis $C$, and then expresses them in terms of the standard basis $B$: For all $\\boldsymbol{v} \\in V$, we have $P_C \\; (\\boldsymbol{v})_C = (\\boldsymbol{v})_B$. The inverse, $P_C^{-1}$, does this the other way around; it takes vectors in terms of the standard basis $B$, and then expresses them in terms of the basis $C$.
\nThe intuition behind the equation $A_T^{B \\rightarrow B} = P_C \\; A_T^{C \\rightarrow C} \\; P_C^{-1}$ is as follows. Looking at the right hand side, $P_C^{-1}$ will take a vector in terms of the standard basis $B$ (say $(\\boldsymbol{v})_B$) and express it in terms of the basis $C$ (i.e. $(\\boldsymbol{v})_C$); $A_T^{C \\rightarrow C}$ will then take $(\\boldsymbol{v})_C$ and give us $T(\\boldsymbol{v})$ in terms of the basis $C$ (i.e. $\\big(T(\\boldsymbol{v}) \\big)_C$); finally $P_C$ takes $\\big(T(\\boldsymbol{v}) \\big)_C$ and expresses it in terms of the standard basis $B$ (i.e. $\\big(T(\\boldsymbol{v}) \\big)_B$)
\nUltimately we have the following: $(\\boldsymbol{v})_B \\longrightarrow (\\boldsymbol{v})_C \\longrightarrow \\big(T(\\boldsymbol{v}) \\big)_C \\longrightarrow \\big(T(\\boldsymbol{v}) \\big)_B$. the overall effect is just $(\\boldsymbol{v})_B \\longrightarrow \\big(T(\\boldsymbol{v}) \\big)_B$, which is the same as applying $A_T^{B \\rightarrow B}$. This is why $A_T^{B \\rightarrow B} = P_C \\; A_T^{C \\rightarrow C} \\; P_C^{-1}$
\n\nNow, let us find $P_C$.
$P_C = \\big( (\\boldsymbol{f_1})_B \\; , \\; (\\boldsymbol{f_2})_B \\big) =$
We can now also find $P_C^{-1}$.
$P_C^{-1} =$
Finally, we have that
$A_T^{B \\rightarrow B} = P_C \\; A_T^{C \\rightarrow C} \\; P_C^{-1} =$
Lent Term Week 3 (lectures 25 and 26): In this question you will look at the transition matrix between two bases, and the matrix of a linear transformation with respect to a non-standard basis.
\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 is the question for Lent Term week 3 of the MA100 course at the LSE. It looks at material from chapters 25 and 26.
"}, "advice": "", "variables": {"P_C": {"definition": "matrix([a , c] , [b , d])", "templateType": "anything", "name": "P_C", "group": "Variables for parts a to d", "description": "This is the transition matrix from basis C to the standard basis B.
"}, "PP_C": {"definition": "matrix([h , k] , [j , l])", "templateType": "anything", "name": "PP_C", "group": "Variables for parts f and g", "description": "This is the transition matrix from basis C to the standard basis B.
"}, "f2": {"definition": "vector(c , d)", "templateType": "anything", "name": "f2", "group": "Variables for parts a to d", "description": "This is our second vector in our basis C.
"}, "f1": {"definition": "vector(a , b)", "templateType": "anything", "name": "f1", "group": "Variables for parts a to d", "description": "This is our first vector in our basis C.
"}, "A_T_BB": {"definition": "PP_C * A_T_CC * PP_C_inv", "templateType": "anything", "name": "A_T_BB", "group": "Variables for parts f and g", "description": "This is the matrix representing the transformation T with respect to the standard basis.
"}, "P_C_inv": {"definition": "(1/(a*d - b*c)) * matrix([d , -c] , [-b , a])", "templateType": "anything", "name": "P_C_inv", "group": "Variables for parts a to d", "description": "This is the inverse of P_C. It is the transition matrix from the standard basis B to basis C.
"}, "w_B": {"definition": "P_C * w_C", "templateType": "anything", "name": "w_B", "group": "Variables for parts a to d", "description": "This is w_C in terms of the basis B.
"}, "c": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "c", "group": "Variables for parts a to d", "description": "This will form part of our second vector in our basis C.
"}, "PP_C_inv": {"definition": "(1/(h*l - k*j)) * matrix([l , -k] , [-j , h])", "templateType": "anything", "name": "PP_C_inv", "group": "Variables for parts f and g", "description": "This is the inverse of PP_C
"}, "a": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "a", "group": "Variables for parts a to d", "description": "This will form part of our first vector in our basis C.
"}, "d": {"definition": "random(-6..6 except(0) except(c*b/a))", "templateType": "anything", "name": "d", "group": "Variables for parts a to d", "description": "This will form part of our second vector in our basis C.
"}, "k": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "k", "group": "Variables for parts f and g", "description": "This is the first entry in the representation of the second vector of the basis C, in terms of the standard basis.
"}, "v_B": {"definition": "vector(1,5)", "templateType": "anything", "name": "v_B", "group": "Variables for parts a to d", "description": "This is avector that we will use in the question. It will be used in terms of the standard basis B.
"}, "j": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "j", "group": "Variables for parts f and g", "description": "This is the second entry in the representation of the first vector of the basis C, in terms of the standard basis.
"}, "w_C": {"definition": "vector(3,4)", "templateType": "anything", "name": "w_C", "group": "Variables for parts a to d", "description": "This is a vector that we will use in the question. It will be used in terms of the basis C.
"}, "e1": {"definition": "vector(1 , 0)", "templateType": "anything", "name": "e1", "group": "Variables for parts a to d", "description": "This is the first vector in the standard basis of R^2. We denote the standard basis by B.
"}, "b": {"definition": "random(-6..6 except(0))", "templateType": "anything", "name": "b", "group": "Variables for parts a to d", "description": "This will form part of our first vector in our basis C.
"}, "h": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "h", "group": "Variables for parts f and g", "description": "This is the first entry in the representation of the first vector of the basis C, in terms of the standard basis.
"}, "f": {"definition": "random(-6..6 except(0) except(1) except(-1))", "templateType": "anything", "name": "f", "group": "Variables for parts f and g", "description": "This will define part of how the transformation T acts on the non-standard basis C.
"}, "e1_C": {"definition": "P_C_inv * e1", "templateType": "anything", "name": "e1_C", "group": "Variables for parts a to d", "description": "This is the vector e1 expressed in terms of the basis C
"}, "v_C": {"definition": "P_C_inv * v_B", "templateType": "anything", "name": "v_C", "group": "Variables for parts a to d", "description": "This is v_B in terms of the basis C.
"}, "e2_C": {"definition": "P_C_inv * e2", "templateType": "anything", "name": "e2_C", "group": "Variables for parts a to d", "description": "This is the vector e2 expressed in terms of the basis C
"}, "g": {"definition": "random(-6..6 except(0) except(1) except(-1) except(f) except(-f))", "templateType": "anything", "name": "g", "group": "Variables for parts f and g", "description": "This will define part of how the transformation T acts on the non-standard basis C.
"}, "e2": {"definition": "vector(0 , 1)", "templateType": "anything", "name": "e2", "group": "Variables for parts a to d", "description": "This is the second vector in the standard basis of R^2. We denote the standard basis by B.
"}, "A_T_CC": {"definition": "matrix([f , 0] , [0 , g])", "templateType": "anything", "name": "A_T_CC", "group": "Variables for parts f and g", "description": "This is the matrix representing the transformation T with respect to the basis C.
"}, "l": {"definition": "random(-6..6 except(0) except(1) except(-1) except(k*j/h))", "templateType": "anything", "name": "l", "group": "Variables for parts f and g", "description": "This is the second entry in the representation of thesecondvector of the basis C, in terms of the standard basis.
"}}, "ungrouped_variables": [], "extensions": [], "functions": {}, "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}