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Jamin2112
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Homework Statement
(Pictured)
Homework Equations
Some Wikipedia and Wolfram MathWorld definitions.
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection.
The rank of a matrix A is the number of linearly independent rows or columns of A.
In linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which Ax = 0.
The nullity of a linear map of vector spaces is the dimension of its null space.
The Attempt at a Solution
I get a little confused with this stuff.
Say that we're looking at the first matrix. It can be row reduced as follows.
So it looks like we know that Ax = 0 whenever
x3 = -1/2 x1
and
x3 = x2.
In other words, any vector x of the form
, where x3 is any real number, will solve Ax = 0. So the above picture is the basis is the null space of A; and since it has one vector, the rank of the null space, the nullity, is 1.
Tell me if I'm understanding this; and if I'm not understanding it, explain it to as if I were a 5-year-old.