2D coordinate grid with standard basis are i=[1,0], j=[0,1]

After changing the basis with example: i=[3,-2], j=[2,1], 2D coordinate grid must look like,

2D coordinate grid with standard basis are i=[1,0], j=[0,1]

After changing the basis with example: i=[3,-2], j=[2,1], 2D coordinate grid must look like,

Hi @lalit,

Welcome to plotly forum! Plotly is a graphics library, but you addressed a matematical question which should be posted on https://math.stackexchange.com/.

It seems to be a homework, and what you want to know is not â€śhow to change basis vector of 2d coordinateâ€ť, but how the linear transformation, T, which maps the standard basis

vectors (i, j), respectively to the vectors v1, v2, given above, acts on the grid lines from your upper plot.

The linear transformation T is represented by its matrix, denoted also T, which has as columns the vectors v1 and v2.

Then T transforms each vector, v, of coordinates [x, y] (with respect to the standard basis), to the vector whose coordinates are the elements of the product T*V:

```
T*[x]
[y]
```

In the following code each vertical line is drawn given the coordinates of its ends and similarly for the horizontal lines.

Then applying the transformation T to the vectors having as coordinates these line ends, you can plot the transformed grid:

```
import plotly.graph_objects as go
import numpy as np
from plotly.subplots import make_subplots
n=m=5
fig=make_subplots(rows=1, cols=2, horizontal_spacing=0.035)
#set up the lists of vertical line x and y-end coordinates
xv = []
yv = []
for k in range(-n, n+1):
xv.extend([k, k, np.nan])
yv.extend([-m, m, np.nan])
lw=1.5 #line_width
fig.add_trace(go.Scatter(x=xv, y=yv, mode="lines", line_width=lw), 1, 1)
#set up the lists of horizontal line x and y-end coordinates
xh=[]
yh=[]
for k in range(-m, m+1):
xh.extend([-m, m, np.nan])
yh.extend([k, k, np.nan])
fig.add_trace(go.Scatter(x=xh, y=yh, mode="lines", line_width=lw), 1, 1)
#define the linear transformation T, such that T(i)=[3, -2]^T, T[j]=[2, 1]^T
T = np.array([[3., 2], [-2, 1.]], dtype=float)
#get only the coordinates from -3 to 3
X = np.array(xv[6:-6])
Y = np.array(yv[6:-6])
# transform by T the vector of coordinates [x, y]^T where the vector runs over the columns of np.stack((X, Y))
Txvyv = T@np.stack((X, Y)) #transform by T the vertical lines
X = np.array(xh[6:-6])
Y = np.array(yh[6:-6])
Txhyh = T@np.stack((X, Y))# #transform by T the horizontal lines
fig.add_trace(go.Scatter(x=Txvyv[0], y=Txvyv[1], mode="lines", line_width=lw), 1, 2)
fig.add_trace(go.Scatter(x=Txhyh[0], y=Txhyh[1], mode="lines", line_width=lw), 1, 2)
fig.update_xaxes(range=[-n, n])
fig.update_yaxes(range=[-m, m])
fig.update_layout(width=800, height=500, showlegend=False, template="none",
plot_bgcolor="black", yaxis2_showgrid=False, xaxis2_showgrid=False)
fig.show()
```

1 Like

Thanks a lot broo

I was right: this was your homework

```
import plotly.graph_objects as go
import numpy as np
from plotly.subplots import make_subplots
n=m=5
#set up the lists of vertical line x and y-end coordinates
xv = []
yv = []
for k in range(-n, n+1):
xv.extend([k, k, np.nan])
yv.extend([-m, m, np.nan])
#set up the lists of horizontal line x and y-end coordinates
xh=[]
yh=[]
for k in range(-m, m+1):
xh.extend([-m, m, np.nan])
yh.extend([k, k, np.nan])
x=np.array(xv+xh)
y=np.array(yv+yh)
# Linear transformation
T = np.array([[3, -2], [2, 1]], dtype=float)
xy=np.stack((x,y))
xy_T=T.T@xy
fig=go.Figure(
data=[
go.Scatter(x=x, y=y, line_width=1),
go.Scatter(x=[0,0.01], y=[0,0],line_width=3),
go.Scatter(x=[0,0], y=[0,0.01],line_width=3)
],
layout=go.Layout(
showlegend=False,
xaxis=dict(range=[-n, n], autorange=False),
yaxis=dict(range=[-m, m], autorange=False),
width=500,
height=500,
updatemenus=[
dict(
type="buttons",
buttons=[dict(label="Play",method="animate",args=[None])]
)
]
),
frames=[
go.Frame(data=[
go.Scatter(x=x, y=y, line_width=1),
go.Scatter(x=[0,0.01], y=[0,0],line_width=3),
go.Scatter(x=[0,0], y=[0,0.01],line_width=3)
]
),
go.Frame(data=[
go.Scatter(x=x, y=y, line_width=1),
go.Scatter(x=[0,1], y=[0,0], line_width=3), # i=[1,0]
go.Scatter(x=[0,0], y=[0,1], line_width=3) # j=[0,1]
]
),
go.Frame(data=[
go.Scatter(x=xy_T[0], y=xy_T[1],line_width=1),
go.Scatter(x=[0,T[0][0]], y=[0,T[0][1]], line_width=3), # i=[3,-2]
go.Scatter(x=[0,T[1][0]], y=[0,T[1][1]], line_width=3), # j=[2,1]
]
),
]
)
fig.show()
```

yepp bro!!