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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 4AEXP

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Question

Which of the following is NOT a subspace of M22?
O
O
W is the set of matrices of the form
and z are any real numbers
W is the set of matrices of the form
x, y, and z are any real numbers
W is the set of matrices of the form
and z are any real numbers
W is the set of matrices of the form
x, y, and z are any real numbers
O None of them
(6).
where x, y,
(2+vő).
(21)
(2723).
, where
where x, y,
where

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Find the bases for the four fundamental subspaces of the matrix. A=[010030101].
Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
A square matrix A=[aij]n with aij=0 for all ij is called upper triangular. Prove or disprove each of the following statements. The set of all upper triangular matrices is closed with respect to matrix addition in Mn(). The set of all upper triangular matrices is closed with respect to matrix multiplication in Mn(). If A and B are square and the product AB is upper triangular, then at least one of A or B is upper triangular.

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