Linear Algebra I & Mathematics Tutorial 1b
Nagoya University, G30 Program, Fall 2023

Instructor: Henrik Bachmann
Teaching assistant: Yuichiro Toma

Homework 6: Linear independence & Basis•Deadline: 14th January, 2024

Exercise 1. (6 Points) Let V ⊂ Rn be a subspace, v1 , . . . , vl ∈ V linearly independent and V =span{w1 , . . . , wm } for some w1 , . . . , wm ∈ Rn . Show that we have l ≤ m. (Without using Lemma 9.4)
In other words: Show that a subspace spanned by m vectors can not contain more than m linearlyindependent vectors.
Exercise 2. (7 Points) Determine bases for the kernel and the image of the following linear map
F : R5 −→ R3


0x 7−→ −1
10−2201−1−21222 x.
−5The following exercise is intended to show the basic idea of 3D computer graphics, by showing how toget a 2-dimensional picture (to be shown on a 2-dimensional monitor) from an 3-dimensional object.
Exercise 3. (7 Points)(i) We deﬁne the corners of a cube with side length 18 in R3 by the following set of 8 points:
 
 w1
W = w2  ∈ R3 | w1 , w2 , w3 ∈ {0, 18} .
w3Make a drawing of a cube with side length 18 in R3 , i.e. draw the 8 points in the set W and connect
two points if they diﬀer just by one entry.
(This just means that you draw a cube like you would usually draw it. ”Diﬀer by one entry” justmeans that these points are on the same edge of the cube.)(ii) Show that D = (d1 , d2 , d3 ) is a basis of R3 , where  −12d1 = 1 , d2 =  1  ,03
3d3 = −6 .
3(iii) Write each x ∈ W as a linear combination in the basis D, i.e. for each x ﬁnd λ1 , λ2 , λ3 ∈ R withx = λ1 d 1 + λ 2 d 2 + λ 3 d 3 .
(iv) For each x ∈ W draw the points (λ1 , λ2 ) in R2 . Connect two points if the corresponding elements
in W just diﬀer by one entry.
Explanation: What you should get in (iv) is a drawing of the 3-dimensional cube in 2 dimensions. Thebasis D somehow describes from which direction you look at the cube. If you replaced the D by thestandard basis (e1 , e2 , e3 ), you would get a picture of the cube from the top (i.e., just a square). The λ3 ,which you did not use for the drawing, describes the distance in the viewing direction.

Version: December 19, 2023

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Merry Christmas, Frohe Weihnachten, メリークリスマス !
Here are today’s words, all of them you’ll ﬁnd on this homework.
せんけいどくりつき ていじげん線形独立基底次元

These words are: senkei dokuritsu (linear independence), kitei (basis) and jigen (dimension).
どくりつ(As we learned in HW1) 線形 means ”linear”.. so naturally 独立 means ”independence”. Note that this
どくりつきねんび

is not only in the context of math; for example, independence day (not the movie!) is 独立記念日.
じげんSimilarly, 次元 here is not only limited to maths; when talking about dimensions in Physics or in everyday
さんじげんに じ げ ん

life, the same word is used. For example, 3D is 三次元 and 2D is 二次元.
せんけい

As 線形 has been discussed in the HW1, it’ll be omitted.
どく独-た-

こくりつ---げん元

しりつThis kanji means ”basic”. It refers to how bases are, in a sense, the ”basicblocks” that one can use to ”create” any element in the subspace. This kanji
きほんてきis commonly used in everyday life in the word 基本的 (In principle).

じ次

ing up” for oneself. Common uses of this kanji include the verb 立ちます (tostand), 国立 (State-established) and 私立 (Private-established).

てい底

どくしん

This kanji means ”to stand”. It refers to how ”independence” implies ”stand-き基

こどくkanji include 孤独 (loneliness) and 独身 (bachelor).

りつ

立This kanji means ”alone”. It refers to how independence means ”standingalone (without being under others’ inﬂuence or control). Other uses of this-This kanji means ”bottom”. This kanji further emphasizes the basic nature
かわそこof bases. While not so common, this kanji is used in 川底 (River bed).
This kanji means ”next”. Usually, this kanji is a stand-alone kanji that one
つぎwould read as 次 (also meaning ”next”).
This kanji means ”origin” or ”former”. While this kanji is not very common in
everyday life, it is used when talking about ”former title holders”, for example”元生徒会長” (Former student council president”).

And that’s it for today. Merry Christmas! Happy new year!
See you again in 2024!

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