01.向量是什麼

05-14

本節主要內容：二維向量－＞三維向量－＞向量的加法－＞向量的數乘．

1.二維向量

提示１：向量的起點可以在任何位置，我們一般假設在原點，向量具有固定長度和方向．

提示２：繪製大量向量時，每一個向量可以用一個點表示（假設起點為原點）．

提示３：注意下圖，向量的橫坐標是-2，縱左邊是3，代表當以原點為起點時，箭頭位置在(-2,3)．在python中，一般用[[-2],[3]]表示．

2．三維向量

提示：三維向量大概長這個樣子，注意右側矩陣的表示形式，是豎著排列的．在python中，一般用[[2],[1],[3]]表示．

3.向量的加法

提示：兩個向量的加法一般將第一個向量末尾作為第二個向量的起點．如下為[[2],[3]]＋[[3],[2]]=[[5],[5]]，因為這裡以原點作為第一個向量的起點，我們可以簡單看到兩個向量之和位於坐標(5,5)．

另一個，按坐標軸拆解的例子：

4.向量的數乘

注意這裡x和y分別為原來的-1.5倍．

-1.5＊[[2],[3]] = [[-3],[-4.5]]．

下一篇地址：

袁傑雄：02.線性組合和張成空間和基<python線性代數>?

zhuanlan.zhihu.com

動畫代碼：

附註代碼１：

import numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.axisartist.axislines import SubplotZerofrom moviepy.editor import VideoClipfrom moviepy.video.io.bindings import mplfig_to_npimagefrom pylab import *zhfont = mpl.font_manager.FontProperties(fname=/usr/share/fonts/truetype/wqy/wqy-microhei.ttc,size=13)plt.rcParams[axes.unicode_minus]=False #負號顯示解決方案plt.style.use("seaborn")def set_axis_middle(fig): ax = SubplotZero(fig, 111) fig.add_subplot(ax) for direction in ["xzero", "yzero"]: # adds arrows at the ends of each axis ax.axis[direction].set_axisline_style("-|>") # adds X and Y-axis from the origin ax.axis[direction].set_visible(True) for direction in ["left", "right", "bottom", "top"]: # hides borders ax.axis[direction].set_visible(False) ax.set_aspect(1.) #設置高寬比 ax.grid(True, zorder=0) ax.set_xlim(-5, 5) ax.set_ylim(-5, 5) return axfig = plt.figure()def update(second): num = second X,Y = -num/2,num/3 ax = set_axis_middle(fig) ax.quiver(X, Y, [-2], [3], angles=xy, scale_units=xy, scale=1) plt.text(X+0.2,Y-0.2,r"$[^{-2}_{3}]$", fontproperties=zhfont) print (num) return mplfig_to_npimage(fig)if 1: animation = VideoClip(update, duration = 2) animation.write_gif("2dvector.gif", fps=10)

附註代碼２：

import numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.axisartist.axislines import SubplotZerofrom moviepy.editor import VideoClipfrom moviepy.video.io.bindings import mplfig_to_npimagefrom pylab import *zhfont = mpl.font_manager.FontProperties(fname=/usr/share/fonts/truetype/wqy/wqy-microhei.ttc,size=13)plt.rcParams[axes.unicode_minus]=False #負號顯示解決方案plt.style.use("seaborn")def set_axis_middle(fig): ax = SubplotZero(fig, 111) fig.add_subplot(ax) for direction in ["xzero", "yzero"]: # adds arrows at the ends of each axis ax.axis[direction].set_axisline_style("-|>") # adds X and Y-axis from the origin ax.axis[direction].set_visible(True) for direction in ["left", "right", "bottom", "top"]: # hides borders ax.axis[direction].set_visible(False) ax.set_aspect(1.) #設置高寬比 ax.grid(True, zorder=0) ax.set_xlim(-0, 5.5) ax.set_ylim(-0, 5.5) return axfig = plt.figure()def draw_vector(ax,x,y,u,v,c): ax.quiver(x,y, u,v,angles=xy, scale_units=xy, scale=1,color = c) plt.text(x+u/2,y+v/2,r"$[^{0}_{1}]$".format(u,v), fontproperties=zhfont)def update(second): num = second U,V = 2, 3 U2,V2 = 3,2 ax = set_axis_middle(fig) draw_vector(ax,0,0,U,V,"y") if second <= 1: X,Y = second*U/1,second*V/1 draw_vector(ax,X,Y,U2,V2,"r") else: draw_vector(ax,U,V,U2,V2,"r") draw_vector(ax,0,0,U+U2,V+V2,"g") print (num) return mplfig_to_npimage(fig)if 1: animation = VideoClip(update, duration = 3) animation.write_gif("test.gif", fps=10)

附註代碼３：

import numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.axisartist.axislines import SubplotZerofrom moviepy.editor import VideoClipfrom moviepy.video.io.bindings import mplfig_to_npimagefrom pylab import *zhfont = mpl.font_manager.FontProperties(fname=/usr/share/fonts/truetype/wqy/wqy-microhei.ttc,size=13)plt.rcParams[axes.unicode_minus]=False #負號顯示解決方案plt.style.use("seaborn")def set_axis_middle(fig): ax = SubplotZero(fig, 111) fig.add_subplot(ax) for direction in ["xzero", "yzero"]: # adds arrows at the ends of each axis ax.axis[direction].set_axisline_style("-|>") # adds X and Y-axis from the origin ax.axis[direction].set_visible(True) for direction in ["left", "right", "bottom", "top"]: # hides borders ax.axis[direction].set_visible(False) ax.set_aspect(1.) #設置高寬比 ax.grid(True, zorder=0) ax.set_xlim(-5, 5) ax.set_ylim(-5, 5) return axfig = plt.figure()def draw_vector(ax,x,y,u,v,c,s="",s2=""): ax.quiver(x,y, u,v,angles=xy, scale_units=xy, scale=1,color = c) if s == "": plt.text(x+u/2,y+v/2,r"$[^{"+str(u)+"}_{"+str(v)+"}]$", fontproperties=zhfont) else: plt.text(x+u/2,y+v/2,s2 + r"$vec{"+s+"}$", fontproperties=zhfont)def update(second): num = second U,V = 2, 3 rate = -1.5 ax = set_axis_middle(fig) if second <= 1: draw_vector(ax,0,0,U,V,"y","v") elif second <= 2: draw_vector(ax,0,0,U,V,"y","v") U2,V2 = (second-1)*U*rate,(second-1)*V*rate draw_vector(ax,0,0,U2,V2,"r","v","{0}".format(rate)) else: second = 2 draw_vector(ax,0,0,U,V,"y") U2,V2 = (second-1)*U*rate,(second-1)*V*rate draw_vector(ax,0,0,U2,V2,"r") print (num) return mplfig_to_npimage(fig)if 1: animation = VideoClip(update, duration = 4) animation.write_gif("test.gif", fps=10)