diff --git a/source/linear-algebra/source/01-LE/02.ptx b/source/linear-algebra/source/01-LE/02.ptx index 2d16ffb70..4191459c3 100644 --- a/source/linear-algebra/source/01-LE/02.ptx +++ b/source/linear-algebra/source/01-LE/02.ptx @@ -859,9 +859,9 @@ rref(A) - + - + format rat diff --git a/source/linear-algebra/source/01-LE/03.ptx b/source/linear-algebra/source/01-LE/03.ptx index 5899d5d9f..aa7c58bc3 100644 --- a/source/linear-algebra/source/01-LE/03.ptx +++ b/source/linear-algebra/source/01-LE/03.ptx @@ -111,7 +111,7 @@ How many solutions must this system have? - + A = [ 3 -2 13 6 2 -2 10 2 @@ -182,7 +182,7 @@ How many solutions must this system have? - + rref([3,-2,13,6;2,-2,10,2;-1,0,-3,1;3,7,0,-2]) @@ -251,7 +251,7 @@ How many solutions must this system have? - + @@ -399,7 +399,7 @@ different solutions. We'll learn how to find such solution sets in - + diff --git a/source/linear-algebra/source/01-LE/04.ptx b/source/linear-algebra/source/01-LE/04.ptx index 7af85e7b3..763cd234e 100644 --- a/source/linear-algebra/source/01-LE/04.ptx +++ b/source/linear-algebra/source/01-LE/04.ptx @@ -188,7 +188,7 @@ in the following set-builder notation. - + diff --git a/source/linear-algebra/source/02-EV/01.ptx b/source/linear-algebra/source/02-EV/01.ptx index cbcd8876b..0f1cfb78c 100644 --- a/source/linear-algebra/source/02-EV/01.ptx +++ b/source/linear-algebra/source/02-EV/01.ptx @@ -182,7 +182,7 @@ we refer to this real number as a scalar.

-

+

Correct the SageMath code cell below to generate an illustration of several vectors belonging to \vspan\left\{\left[\begin{array}{c}1\\2\end{array}\right], @@ -191,43 +191,53 @@ we refer to this real number as a scalar. b\left[\begin{array}{c}-1\\1\end{array}\right]}{a, b \in \IR} in the xy plane.

- + -

+

Based on this illustration, which of these geometrical objects best describes the span of these two vectors? +

+

+ Which of these geometrical objects + best describes the span of these two vectors? +

  1. A line
  2. A plane
  3. A parabola
  4. A circle
-

-

+

Sketch a representation of all the vectors belonging to \vspan\left\{\left[\begin{array}{c}6\\-4\end{array}\right], \left[\begin{array}{c}-3\\2\end{array}\right]\right\} in the xy plane, or adapt the code in the previous activity to illustrate this span.

+

+ Sketch a representation of all the vectors belonging to + \vspan\left\{\left[\begin{array}{c}6\\-4\end{array}\right], + \left[\begin{array}{c}-3\\2\end{array}\right]\right\} + in the xy plane. +

Which of these geometrical objects best describes the span of these two vectors? -

    +

    +
    1. A line
    2. A plane
    3. A parabola
    4. A cube
    -

    @@ -286,7 +296,7 @@ x_3\left[\begin{array}{c}1\\0\\-3\end{array}\right]=0 - +

    @@ -351,7 +361,7 @@ The following are all equivalent statements: --> - + @@ -376,7 +386,7 @@ The following are all equivalent statements: --> - + - diff --git a/source/linear-algebra/source/02-EV/04.ptx b/source/linear-algebra/source/02-EV/04.ptx index 1c7032283..4fc70f884 100644 --- a/source/linear-algebra/source/02-EV/04.ptx +++ b/source/linear-algebra/source/02-EV/04.ptx @@ -319,7 +319,7 @@ which isn't what was asked about. - + @@ -369,7 +369,7 @@ which isn't what was asked about. - + @@ -431,7 +431,7 @@ which isn't what was asked about. - + @@ -474,28 +474,6 @@ which isn't what was asked about.

    - diff --git a/source/linear-algebra/source/02-EV/05.ptx b/source/linear-algebra/source/02-EV/05.ptx index bdef470e9..65f7430eb 100644 --- a/source/linear-algebra/source/02-EV/05.ptx +++ b/source/linear-algebra/source/02-EV/05.ptx @@ -520,7 +520,7 @@ Not a basis, because not only does it fail to span \IR^4, it's also linea - + diff --git a/source/linear-algebra/source/02-EV/06.ptx b/source/linear-algebra/source/02-EV/06.ptx index cfe6e05f2..6bd2a191e 100644 --- a/source/linear-algebra/source/02-EV/06.ptx +++ b/source/linear-algebra/source/02-EV/06.ptx @@ -204,7 +204,7 @@ T=\left\{ - + diff --git a/source/linear-algebra/source/02-EV/07.ptx b/source/linear-algebra/source/02-EV/07.ptx index b97f62ea5..9181bdd55 100644 --- a/source/linear-algebra/source/02-EV/07.ptx +++ b/source/linear-algebra/source/02-EV/07.ptx @@ -196,7 +196,7 @@ is already linearly independent and therefore already a basis for the subspace. - + @@ -238,7 +238,7 @@ Find a basis for this solution space. - + @@ -327,7 +327,7 @@ vector cannot be independent. - + @@ -373,7 +373,7 @@ shape of the data that gets projected onto this single point of the screen? - + diff --git a/source/linear-algebra/source/03-AT/02.ptx b/source/linear-algebra/source/03-AT/02.ptx index 6e214ad0b..6df2c1327 100644 --- a/source/linear-algebra/source/03-AT/02.ptx +++ b/source/linear-algebra/source/03-AT/02.ptx @@ -501,7 +501,7 @@ to the variable v, then compute A*v.

    - + format rat A = [ @@ -553,7 +553,7 @@ T\left(\left[\begin{array}{c} -2 \\ 5 \end{array}\right]\right) - + @@ -574,7 +574,7 @@ T\left(\left[\begin{array}{c} -2 \\ 5 \end{array}\right]\right)

    Compute T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right) using technology.

    - + format rat A = [ diff --git a/source/linear-algebra/source/03-AT/03.ptx b/source/linear-algebra/source/03-AT/03.ptx index bb0bcb58b..37b39accc 100644 --- a/source/linear-algebra/source/03-AT/03.ptx +++ b/source/linear-algebra/source/03-AT/03.ptx @@ -209,7 +209,7 @@ Let T: \IR^3 \rightarrow \IR^2 be the linear transformation with the foll     - + @@ -247,7 +247,7 @@ Find a basis for the kernel of T.

    - + @@ -503,7 +503,7 @@ In general, an arbitrary vector \left[\begin{array}{c}\unknown\\\unknown\\\un - + @@ -589,7 +589,7 @@ Find a basis for the kernel and a basis for the image of T.

    - + @@ -696,7 +696,7 @@ The dimension of the image is called the rank of T (or A< --> - + @@ -770,7 +770,7 @@ Given a matrix M

- + @@ -791,7 +791,7 @@ Given a matrix M

- + @@ -813,7 +813,7 @@ Are the row space and column space of N both equal to \mathbb{R}^3

- + diff --git a/source/linear-algebra/source/03-AT/04.ptx b/source/linear-algebra/source/03-AT/04.ptx index 33609a6ff..76555c1ca 100644 --- a/source/linear-algebra/source/03-AT/04.ptx +++ b/source/linear-algebra/source/03-AT/04.ptx @@ -897,7 +897,7 @@ Which of the following must be true? - + rref([2,1,-1; 4,1,1; 6,2,1]) @@ -921,7 +921,7 @@ Which of the following must be true? - + rref([2,1,-1; 4,1,1; 6,2,0]) @@ -945,7 +945,7 @@ Which of the following must be true? - + rref([2,3;1,-1;1,3]) @@ -969,7 +969,7 @@ Which of the following must be true? - + rref([2,1,-1;4,1,1]) diff --git a/source/linear-algebra/source/04-MX/01.ptx b/source/linear-algebra/source/04-MX/01.ptx index 58a53dfd4..cefa76b2c 100644 --- a/source/linear-algebra/source/04-MX/01.ptx +++ b/source/linear-algebra/source/04-MX/01.ptx @@ -261,7 +261,7 @@ B*A - + B = [3 -4 0 ; 2 0 -1 ; 0 -3 3] A = [2 7 -1 ; 0 3 2 ; 1 1 -1] diff --git a/source/linear-algebra/source/04-MX/02.ptx b/source/linear-algebra/source/04-MX/02.ptx index 14fe9c8eb..59ebf327e 100644 --- a/source/linear-algebra/source/04-MX/02.ptx +++ b/source/linear-algebra/source/04-MX/02.ptx @@ -38,7 +38,7 @@

- + @@ -191,7 +191,7 @@ T^{-1}(\vec e_2)\hspace{1em}T^{-1}(\vec e_3)] - + @@ -251,7 +251,7 @@ Let T\colon\IR^4\to\IR^4 be the linear bijection given by the standard ma - + @@ -278,7 +278,7 @@ Is the matrix \left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 &

- + @@ -316,7 +316,7 @@ Is the matrix \left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 & - + diff --git a/source/linear-algebra/source/04-MX/03.ptx b/source/linear-algebra/source/04-MX/03.ptx index 582acd20e..6608f4e42 100644 --- a/source/linear-algebra/source/04-MX/03.ptx +++ b/source/linear-algebra/source/04-MX/03.ptx @@ -33,7 +33,7 @@

- + @@ -204,7 +204,7 @@

Calculate M_{\mathcal{B}}=B^{-1} using technology.

- + @@ -313,7 +313,7 @@ but where inputs and outputs are all given in \mathcal B-coordinates?

Calculate the matrix M_\mathcal{B}AM_{\mathcal{B}}^{-1}.

- + diff --git a/source/linear-algebra/source/04-MX/04.ptx b/source/linear-algebra/source/04-MX/04.ptx index d7314f6b5..fbb07005c 100644 --- a/source/linear-algebra/source/04-MX/04.ptx +++ b/source/linear-algebra/source/04-MX/04.ptx @@ -184,7 +184,7 @@ the first row when left-multiplying? (R_1+5R_3\to R_1)

- + @@ -313,7 +313,7 @@ Check your work using technology. - + @@ -364,7 +364,7 @@ Check your work using technology. - + diff --git a/source/linear-algebra/source/04-MX/samples/01.ptx b/source/linear-algebra/source/04-MX/samples/01.ptx index c88ac742d..6b7589946 100644 --- a/source/linear-algebra/source/04-MX/samples/01.ptx +++ b/source/linear-algebra/source/04-MX/samples/01.ptx @@ -29,7 +29,7 @@ Of the following three matrices, only two may be multiplied.

Find their product using technology.

- + A = [ -1 3 -2 -3 diff --git a/source/linear-algebra/source/04-MX/samples/04.ptx b/source/linear-algebra/source/04-MX/samples/04.ptx index 462b4f890..e3eb6e439 100644 --- a/source/linear-algebra/source/04-MX/samples/04.ptx +++ b/source/linear-algebra/source/04-MX/samples/04.ptx @@ -53,7 +53,7 @@ \sim\left[\begin{array}{ccc} -5 & 15 & 15 \\ 2 & -6 & -5 \\ -4 & 12 & 12 \end{array}\right] \sim\left[\begin{array}{ccc} -4 & 12 & 12 \\ 2 & -6 & -5 \\ -5 & 15 & 15 \end{array}\right]

- + C = [ -5 0 0 diff --git a/source/linear-algebra/source/05-GT/02.ptx b/source/linear-algebra/source/05-GT/02.ptx index 4fd315b0b..1b65303c7 100644 --- a/source/linear-algebra/source/05-GT/02.ptx +++ b/source/linear-algebra/source/05-GT/02.ptx @@ -411,7 +411,7 @@ Based on the previous activities, which technique is easier for computing determ -You can check your answers using technology. +

You can check your answers using technology.

det([4,-3,0,0; 1,-3,2,-1; 3,2,0,3; 0,-3,2,-2]) diff --git a/source/linear-algebra/source/05-GT/04.ptx b/source/linear-algebra/source/05-GT/04.ptx index cd38ec074..b833b903e 100644 --- a/source/linear-algebra/source/05-GT/04.ptx +++ b/source/linear-algebra/source/05-GT/04.ptx @@ -52,7 +52,7 @@ to find all the eigenvectors \vec x such that A\vec x=-2\vec x. - + @@ -80,7 +80,7 @@ associated with the eigenvalue 3. - + @@ -97,7 +97,7 @@ associated with the eigenvalue 1. - + @@ -116,7 +116,7 @@ associated with the eigenvalue 2. - +