Mark the following either true or false. Justification is required. (a) The dimensions of Col A and Nul A add up to the number of rows of A. (b) If B is a basis of subspace H, then each vector in H can be written in one and only one way as a linear combination of the vectors in B. (c) The dimension of Nul A is the number of free variables in the equa- tion Az = 0.
Mark the following either true or false. Justification is required. (a) The dimensions of Col A and Nul A add up to the number of rows of A. (b) If B is a basis of subspace H, then each vector in H can be written in one and only one way as a linear combination of the vectors in B. (c) The dimension of Nul A is the number of free variables in the equa- tion Az = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Mark the following either true or false. Justification is required.
(a) The dimensions of \(\text{Col } A\) and \(\text{Nul } A\) add up to the number of rows of \(A\).
(b) If \(\mathcal{B}\) is a basis of subspace \(H\), then each vector in \(H\) can be written in one and only one way as a linear combination of the vectors in \(\mathcal{B}\).
(c) The dimension of \(\text{Nul } A\) is the number of free variables in the equation \(Ax = 0\).
Expert Solution
Step 1: Formula used
To answer the questions we can use following formula
(a) For a matrix A with order m × n,
Rank (A) + Nullity (A) = Number of columns in A
(b) Elements of a basis vectors are linearly independent.
(c) Using rank nullity theorem,
number of columns of A - Rank of A = nullity of A
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