Subspace (reveal example) Let \(W\) be the subset of \(\mathbb{R}^3\) made from the straight line through the origin and \( ({@p1[1]@}, {@p1[2]@}, {@p1[3]@}) \). Is \(W\) a subspace of \(\mathbb{R}^3\)?

True or false: \(W\) is a subspace? [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]

[[reveal input="ans1" value="true"]]

If true write the subspace in parametric form: [[input:ans2]] where the list of parameters [[input:ans3]] \(\in \mathbb{R}\).
[[validation:ans2]] [[validation:ans3]] [[feedback:prt2]]

[[/reveal]] [[reveal input="ans1" value="false"]]

If false the give reasons below.

True or false: \(W\) does not contain the origin (and so is not a subspace)? [[input:ans0]] [[validation:ans0]] [[feedback:prt3]]
\(W\) is not a subspace because it is not closed under addition. For example, \(v_1 = \) [[input:ansv1]] [[validation:ansv1]] and \(v_2 = \) [[input:ansv2]] [[validation:ansv2]] are in \(W\) but their sum is not. [[feedback:prt4]]
\(W\) is not a subspace because it is not closed under scalar multiplication. For example, \(w_1 = \) [[input:answ1]] [[validation:answ1]] is in \(W\) and \(\lambda = \) [[input:answ2]] [[validation:answ2]] is a scalar but the scalar multiple \(\lambda w_1\) is not in \(W\). [[feedback:prt5]]

[[/reveal]]]]> A line through the origin is a subspace. The subspace can be described in parametric form with a single parameter, multiplying the position vector of any non-zero point on the line. 3 0.1 0 2023051500 p1:[-1,3,2]; tlv:[t]; issubspace:true; iszero:true; /* This is the parametric form of W. */ Ws:transpose(matrix(t*p1)); /* Decide if a vector is in W */ find_point_in_space(W, wx):= linsolve(flatten(args(W))-first(args(transpose(wx))), listofvars(W)); point_in_space(W, wx):=not(emptyp(find_point_in_space(W, wx))); /* Calculate the canonical form of a column space of a subspace. */ cspace(ex):= ev(transpose(rref(transpose(coefmatrix(flatten(args(ex)), listofvars(ex))))), simp); /* Is zero in the teacher's answer? */ iszeroreally:not(emptyp(point_in_space(Ws,matrix([0],[0],[0])))); /* Counter examples to addition */ tav1:transpose(matrix([null,null,null])); tav2:transpose(matrix([null,null,null])); /* Counter examples to multiplication */ taw1:transpose(matrix([null,null,null])); taw2:null; {@Ws@} 1 0 0 Correct answer, well done.]]> Your answer is partially correct.]]> Incorrect answer.]]> none 1 i cos-1 lang [ ans0 boolean EMPTYANSWER 15 1 0 0 1 0 0 0 0 ans1 boolean issubspace 15 1 0 0 1 0 0 0 0 ans2 matrix Ws 5 1 5 0 1 0 0 1 1 ans3 algebraic tlv 5 1 5 [?, ...] 0 1 0 1 0 0 ansv1 matrix tav1 2 1 5 0 1 0 0 0 0 ansv2 matrix tav1 2 1 5 0 1 0 0 0 0 answ1 matrix tav1 3 1 5 0 1 0 0 0 0 answ2 numerical EMPTYANSWER 3 1 5 0 1 0 0 0 0 prt1 1.0000000 1 1 0 True false check. AlgEquiv ans1 issubspace 0 = 1 -1 prt1-1-T You have correctly identified if \(W\) is a subspace. = 0 -1 prt1-1-F You have not correctly identified if \(W\) is a subspace. prt2 2.0000000 1 1 lv:listofvars(ans2); deg:apply(max, maplist(lambda([ex], hipow(ans2, ex)), lv)); 0 AlgEquiv ans1 true 0 = 0 1 prt2-1-T = 0 0 -1 prt2-1-F You can only answer this part if you think \(W\) is a subspace! 1 AlgEquiv deg 1 1 + 0 2 prt2-2-T - 0 -1 prt2-2-F The degree of you system is not equal to 1. 2 AlgEquiv point_in_space(ans2, matrix([0],[0],[0])) true 1 + 0 3 prt2-3-T - 0 -1 prt2-3-F Your system does not pass through the origin, but should do! 3 AlgEquiv cspace(ans2) cspace(Ws) 1 = 1 -1 prt2-4-T - 0 -1 prt2-4-F prt3 0.0000000 1 1 0 AlgEquiv ans1 false 0 = 0 1 prt3-1-T = 0 0 -1 prt3-1-F You can only answer this part if you think \(W\) is not a subspace! 1 AlgEquiv ans0 not(point_in_space(Ws,matrix([0],[0],[0]))) 0 = 0 -1 prt3-2-T You have correctly identified whether the origin is in \(W\). = 0 -1 prt3-2-F You have incorrectly identified whether the origin is \(W\). prt4 0.0000000 1 1 0 AlgEquiv ans1 false 0 = 0 1 prt4-1-T = 0 -1 prt4-1-F You can only answer this part if you think \(W\) is not a subspace! 1 AlgEquiv point_in_space(Ws,ansv1) true 0 + 0 2 prt4-2-T - 0 -1 prt4-2-F You are claiming that \({@ansv1@}\) is in \(W\) but it is not! 2 AlgEquiv point_in_space(Ws,ansv2) true 0 = 0 3 prt4-3-T - 0 -1 prt4-3-F You are claiming that \({@ansv2@}\) is in \(W\) but it is not! 3 AlgEquiv point_in_space(Ws,ansv1+ansv2) false 0 = 1 -1 prt4-4-T The point \[ {@ansv1@}+{@ansv2@} = {@ansv1+ansv2@} \not \in W\] as claimed. Hence \(W\) is not a subspace - 0 -1 prt4-4-F The point \[ {@ansv1@}+{@ansv2@} = {@ansv1+ansv2@} \in W.\] Hence your points do not contradict closure of \(W\) under addition. prt5 0.0000000 1 1 0 AlgEquiv ans1 false 0 = 0 1 prt5-1-T = 0 -1 prt5-1-F You can only answer this part if you think \(W\) is not a subspace! 1 AlgEquiv point_in_space(Ws,answ1) true 0 + 0 2 prt5-2-T - 0 -1 prt5-2-F You are claiming that \({@answ1@}\) is in \(W\) but it is not! 2 AlgEquiv point_in_space(Ws,answ2*answ1) false 0 = 1 -1 prt5-3-T The point \[ {@answ2@}\times{@answ1@} = {@answ2*answ1@} not \in W.\] Hence your points do contradict closure of \(W\) under scalar multiplication. Hence \(W\) is not a subspace - 0 -1 prt5-3-F The point \[ {@answ2@}\times{@answ1@} = {@answ2*answ1@} \in W.\] Hence your points to not contradict closure of \(W\) under scalar multiplication. 1 ans0 ans1 issubspace ans2 Ws ans3 tlv ansv1 ansv2 answ1 answ2 prt1 1.0000000 0.0000000 prt1-1-T prt2 1.0000000 0.0000000 prt2-4-T prt3 NULL prt4 NULL prt5 NULL 2 ans0 ans1 issubspace ans2 ev(transpose(matrix(-1*s*p1)),simp) ans3 [s] ansv1 ansv2 answ1 answ2 prt1 1.0000000 0.0000000 prt1-1-T prt2 1.0000000 0.0000000 prt2-4-T prt3 NULL prt4 NULL prt5 NULL 3 ans0 ans1 issubspace ans2 ev(transpose(matrix(t*(p1+[1,0,0]))),simp) ans3 [t] ansv1 ansv2 answ1 answ2 prt1 1.0000000 0.0000000 prt1-1-T prt2 0.0000000 prt2-4-F prt3 NULL prt4 NULL prt5 NULL 4 ans0 true ans1 false ans2 ans3 [?, ...] ansv1 ansv2 answ1 answ2 prt1 0.0000000 0.1000000 prt1-1-F prt2 NULL prt3 0.0000000 0.1000000 prt3-2-F prt4 NULL prt5 NULL 5 ans0 false ans1 false ans2 ans3 [?, ...] ansv1 ansv2 answ1 answ2 prt1 0.0000000 0.1000000 prt1-1-F prt2 NULL prt3 0.0000000 0.1000000 prt3-2-T 6 ans0 ans1 false ans2 ans3 [?, ...] ansv1 ev(transpose(matrix(p1)),simp) ansv2 ev(transpose(matrix(-2*p1)), simp) answ1 ev(transpose(matrix(p1)),simp) answ2 3 prt1 0.0000000 0.1000000 prt1-1-F prt2 NULL prt3 NULL prt4 0.0000000 0.1000000 prt4-4-F prt5 0.0000000 0.1000000 prt5-3-F 7 Student chooses the wrong answer ans0 true ans1 false ans2 ans3 [?, ...] ansv1 ansv2 answ1 answ2 prt1 0.0000000 prt1-1-F prt2 NULL prt3 0.0000000 prt3-2-F prt4 NULL prt5 NULL