What kind of matrices can be visualized by graphing the column vectors?
$begingroup$
In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis vectors. Picture of the grid.
These basis vectors $hat i=<3,-2>$ and $hat j=<2,1>$ can be put in a $2times 2$ transformation matrix where they can apply the same transformation to any $Bbb R^2$ vector.
$$T(vec x)=begin{bmatrix}3&2\-2&1end{bmatrix}vec x$$
What kind of transformation matrices (besides one using a $2times 2$ matrix) can be visualized by making a grid based off the column vectors. For example, is it possible to create a grid off the following transformation?
$$T(vec x)=begin{bmatrix}3&2&2\-2&1&0end{bmatrix}vec x$$
Or is creating a grid only possible if I know that the columns of the matrix correspond to basis vectors?
linear-algebra matrices visualization
edited Jan 16 at 19:54
Mutantoe
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
$endgroup$
add a comment |
$begingroup$
In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis vectors. Picture of the grid.
These basis vectors $hat i=<3,-2>$ and $hat j=<2,1>$ can be put in a $2times 2$ transformation matrix where they can apply the same transformation to any $Bbb R^2$ vector.
$$T(vec x)=begin{bmatrix}3&2\-2&1end{bmatrix}vec x$$
What kind of transformation matrices (besides one using a $2times 2$ matrix) can be visualized by making a grid based off the column vectors. For example, is it possible to create a grid off the following transformation?
$$T(vec x)=begin{bmatrix}3&2&2\-2&1&0end{bmatrix}vec x$$
Or is creating a grid only possible if I know that the columns of the matrix correspond to basis vectors?
linear-algebra matrices visualization
edited Jan 16 at 19:54
Mutantoe
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
$endgroup$
1
$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51
add a comment |
$begingroup$
In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis vectors. Picture of the grid.
These basis vectors $hat i=<3,-2>$ and $hat j=<2,1>$ can be put in a $2times 2$ transformation matrix where they can apply the same transformation to any $Bbb R^2$ vector.
$$T(vec x)=begin{bmatrix}3&2\-2&1end{bmatrix}vec x$$
What kind of transformation matrices (besides one using a $2times 2$ matrix) can be visualized by making a grid based off the column vectors. For example, is it possible to create a grid off the following transformation?
$$T(vec x)=begin{bmatrix}3&2&2\-2&1&0end{bmatrix}vec x$$
Or is creating a grid only possible if I know that the columns of the matrix correspond to basis vectors?
linear-algebra matrices visualization
edited Jan 16 at 19:54
Mutantoe
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
$endgroup$
In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis vectors. Picture of the grid.
These basis vectors $hat i=<3,-2>$ and $hat j=<2,1>$ can be put in a $2times 2$ transformation matrix where they can apply the same transformation to any $Bbb R^2$ vector.
$$T(vec x)=begin{bmatrix}3&2\-2&1end{bmatrix}vec x$$
What kind of transformation matrices (besides one using a $2times 2$ matrix) can be visualized by making a grid based off the column vectors. For example, is it possible to create a grid off the following transformation?
$$T(vec x)=begin{bmatrix}3&2&2\-2&1&0end{bmatrix}vec x$$
Or is creating a grid only possible if I know that the columns of the matrix correspond to basis vectors?
linear-algebra matrices visualization
linear-algebra matrices visualization
edited Jan 16 at 19:54
Mutantoe
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
edited Jan 16 at 19:54
Mutantoe
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
edited Jan 16 at 19:54
Mutantoe
619513
edited Jan 16 at 19:54
Mutantoe
619513
edited Jan 16 at 19:54
Mutantoe
619513
619513
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
asked Jan 16 at 19:27
mrhumanzeemrhumanzee
12
12
1
$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51
add a comment |
1
$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51
1
1
$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51
add a comment |
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$begingroup$
This is true for any matrix since this is a visualization of a linear map. However, in dimensions bigger than 2 or 3 its not really possible to easily visualize it. You can think of $T(x)$ as picking a linear combination of the columns of $A$ where the coefficients are from $x$.
$endgroup$
– tch
Jan 16 at 19:50
$begingroup$
How do I visualize a grid for the transformation matrix with the column $<2,0>$?
$endgroup$
– mrhumanzee
Jan 16 at 20:11
$begingroup$
You make the grid by drawing all three vectors, and from each tip drawing all three again. This represents a map from $mathbb{R}^3$ to $mathbb{R}^2$ so you can think of it as "squashing" the uniform grid you would get from the vectors $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $mathbb{R}^3$ to a 2d plane.
$endgroup$
– tch
Jan 16 at 20:51