Linear algebra - null spaces. Please do a) and b) and possibly explain why? (12)

Question

Linear algebra - null spaces. Please do a) and b) and possibly explain why? (12) Sometimes/always/never. Circle the word (letter) that completes the statement. A, then the columns of A ares N linearly independent. X (a) If A is a matrix and 0 is in N (b) If A is a matrix and is in Nul A, then the columns of A ardS A N linearly indep (c) If an, a2, a3, and as are vectors in R3, then a2, a3, a4) is S A a basis for R3 a1, (d) If a,az,aa, and an are vectors in R4, then la1, a2, a3,a4 is S A Nabasis for Ri.

Explanation / Answer

We have a theorem which states that "Null Space of a matrix A Contains Only Zero Vector iff Columns of the matrix A are Independent"

(a) Some times ( if there is no other vector in the null space of A then only we can confidently say that the columns are linearly independent)

(b) Never (because null space contains a non-zero vector, by the above theorem the columns of A can not be linearly independent)

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