Definition 3.6.4. The kernel of a matrix A is the set ker(A) = {v |A· v = 0}. Problem 2: Using the same generality (i.e. abstractly, not using a particular matrix) as above, prove that if u and v are both in the kernel of a matrix A, then Span{u, v} C ker(A). Does it change anything if I take 3 vectors? 4 vectors? k-many vectors?
Definition 3.6.4. The kernel of a matrix A is the set ker(A) = {v |A· v = 0}. Problem 2: Using the same generality (i.e. abstractly, not using a particular matrix) as above, prove that if u and v are both in the kernel of a matrix A, then Span{u, v} C ker(A). Does it change anything if I take 3 vectors? 4 vectors? k-many vectors?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve Problem 2. Thank you
Transcribed Image Text:Problem 1: Consider the matrix B =
so that BC = CB.
The proof of theorem 3.2.5 depends on how one approaches the subject: it can either be a definition or a theorem. But
for us, one would have to write out an arbitrary matrix, and check the properties. From there, we can deduce an additional
property as explained below:
1
Theorem 3.6.3. If A E Matmxn and 0 =
Theorem 3.2.5. If A E Matmxn, u, v E R", and a € R,
then A(u + v) = Au + Av and A(au) = a(Au).
Proof.
Solve for all matrices C =
ER then A. 0 = 0 € Rm
с
0=(A.0) + A-Othm.3.2.5 A (-0) + A⋅ (0) thm.3.2.5 A-(-0+0) = A.0.
Definition 3.6.4. The kernel of a matrix A is the set ker(A) = {v |A·v=0}.
Problem 2: Using the same generality (i.e. abstractly, not using a particular matrix) as above, prove that if u and v are
both in the kernel of a matrix A, then Span{u, v} C ker(A). Does it change anything if I take 3 vectors? 4 vectors? k-many
vectors?
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