Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 22E from Chapter 4.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 22E
Chapter:
Problem:
Step-by-Step Solution
Given Information
We are given with a fixed vector F having dimensions $3 \times 2$. H is a set of all matrices A in ${M_{2 \times 4}}$ such that $FA=0$. We have to determine if H is a subspace of ${M_{2 \times 4}}$
Step-1: Properties of a subspace
A set is a subspace of the vector space if it satisfies the following properties.
(a) The zero vector is in H
(b) The each u and v in H, the sum of u+v is in H
(c) For each u in H and a scalar c, the vector $cu$ is in H
Step-2: Check for zero vector
Let the zero matrix O is in ${M_{2 \times 4}}$, then \[FO = 0\] therefore the zero vector is in H.
Step-3: Closure under addition
If, A and B are in H, then check if their addition is also in H. \[\begin{array}{l} F\left( {A + B} \right) = FA + FB\\ \\ F\left( {A + B} \right) = 0 + 0\\ \\ F\left( {A + B} \right) = 0 \end{array}\] Hence A+B is in H .
Step-4: Scalar Multiplication
Let c be any scalar and A is in H, then \[\begin{array}{l} F\left( {cA} \right) = cFA\\ F\left( {cA} \right) = c \times 0 + 0\\ F\left( {cA} \right) = 0 \end{array}\] Hence cA is in H .
H is a subspace of ${M_{2 \times 4}}$
We are given with a fixed vector F having dimensions $3 \times 2$. H is a set of all matrices A in ${M_{2 \times 4}}$ such that $FA=0$. We have to determine if H is a subspace of ${M_{2 \times 4}}$
Step-1: Properties of a subspace
A set is a subspace of the vector space if it satisfies the following properties.
(a) The zero vector is in H
(b) The each u and v in H, the sum of u+v is in H
(c) For each u in H and a scalar c, the vector $cu$ is in H
Step-2: Check for zero vector
Let the zero matrix O is in ${M_{2 \times 4}}$, then \[FO = 0\] therefore the zero vector is in H.
Step-3: Closure under addition
If, A and B are in H, then check if their addition is also in H. \[\begin{array}{l} F\left( {A + B} \right) = FA + FB\\ \\ F\left( {A + B} \right) = 0 + 0\\ \\ F\left( {A + B} \right) = 0 \end{array}\] Hence A+B is in H .
Step-4: Scalar Multiplication
Let c be any scalar and A is in H, then \[\begin{array}{l} F\left( {cA} \right) = cFA\\ F\left( {cA} \right) = c \times 0 + 0\\ F\left( {cA} \right) = 0 \end{array}\] Hence cA is in H .
H is a subspace of ${M_{2 \times 4}}$