Introduction to Linear Algebra, Fifth Edition
5th Edition
ISBN: 9780980232776
Author: Gilbert Strang
Publisher: Wellesley-Cambridge Press
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Chapter 3.4, Problem 29PS
To determine
(a)
The subspace of
The invertible matrices span the space of all
Given:
Calculation:
It is noted that a set of vectors spans a whole space if their linear combinations fill the space.
Therefore, the invertible matrices span the space of all
(b)
The subspace of
The rank one matrix span the space of all
Given:
Calculation:
It is noted that rank of a matrix is its number of pivots. If the rank one matrix has only one pivot, it means that its
Therefore, the rank one matrix spans the space of all
(c)
The subspace of
I itself spans the space of all multiples
Given:
Calculation:
It is noted that the identity matrix itself spans the space of all matrices.
Therefore, I by itself spans the space of all multiples
To determine
(b)
The subspace of
The rank one matrix span the space of all
Given:
Calculation:
It is noted that rank of a matrix is its number of pivots. If the rank one matrix has only one pivot, it means that its
Therefore, the rank one matrix spans the space of all
(c)
The subspace of
I itself spans the space of all multiples
Given:
Calculation:
It is noted that the identity matrix itself spans the space of all matrices.
Therefore, I by itself spans the space of all multiples
To determine
(c)
The subspace of
I itself spans the space of all multiples
Given:
Calculation:
It is noted that the identity matrix itself spans the space of all matrices.
Therefore, I by itself spans the space of all multiples
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Which statement is true about the graph of the equation y = csc¯¹(x)?
There is a horizontal asymptote at y = 0.
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There is a horizontal asymptote at y = 2.
There is a vertical asymptote at x = 0.
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ے ملزمة احمد
Q (a) Let f be a linear map from a space X into a space Y and (X1,X2,...,xn) basis for X, show that fis one-to-
one iff (f(x1),f(x2),...,f(x) } linearly independent.
(b) Let X= {ao+ax₁+a2x2+...+anxn, a;ER} be a vector space over R, write with prove a hyperspace and a
hyperplane of X.
مبر خد احمد
Q₂ (a) Let M be a subspace of a vector space X, and A= {fex/ f(x)=0, x E M ), show that whether A is
convex set or not, affine set or not.
Write with prove an
application of Hahn-Banach theorem.
Show that every singleton set in a normed space X is closed and any finite set in X is closed (14M)
Let M be a proper subspace of a finite dimension vector space X over a field F show that
whether: (1) If S is a base for M then S base for X or not, (2) If T base for X then base for M
or not.
(b) Let X-P₂(x) be a vector space over polynomials a field of real numbers R, write with L
prove convex subset of X and hyperspace of X.
Q₂/ (a) Let X-R³ be a vector space over a over a field of real numbers R and
A=((a,b,o), a,bE R), A is a subspace of X, let g be a function from A into R such that
gla,b,o)-a, gEA, find fe X such that g(t)=f(t), tEA.
(b) Let M be a non-empty subset of a space X, show that M is a hyperplane of X iff there
Xiff there
exists fE X/10) and tE F such that M=(xE X/ f(x)=t).
(c) Show that the relation equivalent is an equivalence relation on set of norms on a space
X.
Chapter 3 Solutions
Introduction to Linear Algebra, Fifth Edition
Ch. 3.1 - Prob. 1PSCh. 3.1 - Prob. 2PSCh. 3.1 - Prob. 3PSCh. 3.1 - Prob. 4PSCh. 3.1 - Prob. 5PSCh. 3.1 - Prob. 6PSCh. 3.1 - Prob. 7PSCh. 3.1 - Prob. 8PSCh. 3.1 - Prob. 9PSCh. 3.1 - Prob. 10PS
Ch. 3.1 - Prob. 11PSCh. 3.1 - Prob. 12PSCh. 3.1 - Prob. 13PSCh. 3.1 - Prob. 14PSCh. 3.1 - Prob. 15PSCh. 3.1 - Prob. 16PSCh. 3.1 - Prob. 17PSCh. 3.1 - Prob. 18PSCh. 3.1 - Prob. 19PSCh. 3.1 - Prob. 20PSCh. 3.1 - Prob. 21PSCh. 3.1 - Prob. 22PSCh. 3.1 - Prob. 23PSCh. 3.1 - Prob. 24PSCh. 3.1 - Prob. 25PSCh. 3.1 - Prob. 26PSCh. 3.1 - Prob. 27PSCh. 3.1 - Prob. 28PSCh. 3.1 - Prob. 29PSCh. 3.1 - Prob. 30PSCh. 3.1 - Prob. 31PSCh. 3.1 - Prob. 32PSCh. 3.2 - Prob. 1PSCh. 3.2 - Prob. 2PSCh. 3.2 - Prob. 3PSCh. 3.2 - Prob. 4PSCh. 3.2 - Prob. 5PSCh. 3.2 - Prob. 6PSCh. 3.2 - Prob. 7PSCh. 3.2 - Prob. 8PSCh. 3.2 - Prob. 9PSCh. 3.2 - Prob. 10PSCh. 3.2 - Prob. 11PSCh. 3.2 - Prob. 12PSCh. 3.2 - Prob. 13PSCh. 3.2 - Prob. 14PSCh. 3.2 - Prob. 15PSCh. 3.2 - Prob. 16PSCh. 3.2 - Prob. 17PSCh. 3.2 - Prob. 18PSCh. 3.2 - Prob. 19PSCh. 3.2 - Prob. 20PSCh. 3.2 - Prob. 21PSCh. 3.2 - Prob. 22PSCh. 3.2 - Prob. 23PSCh. 3.2 - Prob. 24PSCh. 3.2 - Prob. 25PSCh. 3.2 - Prob. 26PSCh. 3.2 - Prob. 27PSCh. 3.2 - Prob. 28PSCh. 3.2 - Prob. 29PSCh. 3.2 - Prob. 30PSCh. 3.2 - Prob. 31PSCh. 3.2 - Prob. 32PSCh. 3.2 - Prob. 33PSCh. 3.2 - Prob. 34PSCh. 3.2 - Prob. 35PSCh. 3.2 - Prob. 36PSCh. 3.2 - Prob. 37PSCh. 3.2 - Prob. 38PSCh. 3.2 - Prob. 39PSCh. 3.2 - Prob. 40PSCh. 3.2 - Prob. 41PSCh. 3.2 - Prob. 42PSCh. 3.2 - Prob. 43PSCh. 3.2 - Prob. 44PSCh. 3.2 - Prob. 45PSCh. 3.2 - Prob. 46PSCh. 3.2 - Prob. 47PSCh. 3.2 - Prob. 48PSCh. 3.2 - Prob. 49PSCh. 3.2 - Prob. 50PSCh. 3.2 - Prob. 51PSCh. 3.2 - Prob. 52PSCh. 3.2 - Prob. 53PSCh. 3.2 - Prob. 54PSCh. 3.2 - Prob. 55PSCh. 3.2 - Prob. 56PSCh. 3.2 - Prob. 57PSCh. 3.2 - Prob. 58PSCh. 3.2 - Prob. 59PSCh. 3.2 - Prob. 60PSCh. 3.3 - Prob. 1PSCh. 3.3 - Prob. 2PSCh. 3.3 - Prob. 3PSCh. 3.3 - Prob. 4PSCh. 3.3 - Prob. 5PSCh. 3.3 - Prob. 6PSCh. 3.3 - Prob. 7PSCh. 3.3 - Prob. 8PSCh. 3.3 - Prob. 9PSCh. 3.3 - Prob. 10PSCh. 3.3 - Prob. 11PSCh. 3.3 - Prob. 12PSCh. 3.3 - Prob. 13PSCh. 3.3 - Prob. 14PSCh. 3.3 - Prob. 15PSCh. 3.3 - Prob. 16PSCh. 3.3 - Prob. 17PSCh. 3.3 - Prob. 18PSCh. 3.3 - Prob. 19PSCh. 3.3 - Prob. 20PSCh. 3.3 - Prob. 21PSCh. 3.3 - Prob. 22PSCh. 3.3 - Prob. 23PSCh. 3.3 - Prob. 24PSCh. 3.3 - Prob. 25PSCh. 3.3 - Prob. 26PSCh. 3.3 - Prob. 27PSCh. 3.3 - Prob. 28PSCh. 3.3 - Prob. 29PSCh. 3.3 - Prob. 30PSCh. 3.3 - Prob. 31PSCh. 3.3 - Prob. 32PSCh. 3.3 - Prob. 33PSCh. 3.3 - Prob. 34PSCh. 3.3 - Prob. 35PSCh. 3.3 - Prob. 36PSCh. 3.3 - Prob. 37PSCh. 3.4 - Prob. 1PSCh. 3.4 - Prob. 2PSCh. 3.4 - Prob. 3PSCh. 3.4 - Prob. 4PSCh. 3.4 - Prob. 5PSCh. 3.4 - Prob. 6PSCh. 3.4 - Prob. 7PSCh. 3.4 - Prob. 8PSCh. 3.4 - Prob. 9PSCh. 3.4 - Prob. 10PSCh. 3.4 - Prob. 11PSCh. 3.4 - Prob. 12PSCh. 3.4 - Prob. 13PSCh. 3.4 - Prob. 14PSCh. 3.4 - Prob. 15PSCh. 3.4 - Prob. 16PSCh. 3.4 - Prob. 17PSCh. 3.4 - Prob. 18PSCh. 3.4 - Prob. 19PSCh. 3.4 - Prob. 20PSCh. 3.4 - Prob. 21PSCh. 3.4 - Prob. 22PSCh. 3.4 - Prob. 23PSCh. 3.4 - Prob. 24PSCh. 3.4 - Prob. 25PSCh. 3.4 - Prob. 26PSCh. 3.4 - Prob. 27PSCh. 3.4 - Prob. 28PSCh. 3.4 - Prob. 29PSCh. 3.4 - Prob. 30PSCh. 3.4 - Prob. 31PSCh. 3.4 - Prob. 32PSCh. 3.4 - Prob. 33PSCh. 3.4 - Prob. 34PSCh. 3.4 - Prob. 35PSCh. 3.4 - Prob. 36PSCh. 3.4 - Prob. 37PSCh. 3.4 - Prob. 38PSCh. 3.4 - Prob. 39PSCh. 3.4 - Prob. 40PSCh. 3.4 - Prob. 41PSCh. 3.4 - Prob. 42PSCh. 3.4 - Prob. 43PSCh. 3.4 - Prob. 44PSCh. 3.4 - Prob. 45PSCh. 3.4 - Prob. 46PSCh. 3.5 - Prob. 1PSCh. 3.5 - Prob. 2PSCh. 3.5 - Prob. 3PSCh. 3.5 - Prob. 4PSCh. 3.5 - Prob. 5PSCh. 3.5 - Prob. 6PSCh. 3.5 - Prob. 7PSCh. 3.5 - Prob. 8PSCh. 3.5 - Prob. 9PSCh. 3.5 - Prob. 10PSCh. 3.5 - Prob. 11PSCh. 3.5 - Prob. 12PSCh. 3.5 - Prob. 13PSCh. 3.5 - Prob. 14PSCh. 3.5 - Prob. 15PSCh. 3.5 - Prob. 16PSCh. 3.5 - Prob. 17PSCh. 3.5 - Prob. 18PSCh. 3.5 - Prob. 19PSCh. 3.5 - Prob. 20PSCh. 3.5 - Prob. 21PSCh. 3.5 - Prob. 22PSCh. 3.5 - Prob. 23PSCh. 3.5 - Prob. 24PSCh. 3.5 - Prob. 25PSCh. 3.5 - Prob. 26PSCh. 3.5 - Prob. 27PSCh. 3.5 - Prob. 28PSCh. 3.5 - Prob. 29PSCh. 3.5 - Prob. 30PSCh. 3.5 - Prob. 31PS
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