Math 214 Linear Algebra

Non-homework questions that may appear on exams

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Mid 3:

1. Suppose matrix B is obtained from matrix A by one elementary row operation. Prove that column k of A is a linear combination of the other columns of A iff column k of B is a linear combination of the other columns of B.
2. Prove Theorem 4.3a,c-f.
3. Prove Theorem 4.22, parts a, b, d, e.
4. Prove Theorems 5.1 and 5.2 .

Mid 2:

1. Explain why b is in the span of the columns of A iff the system [A | b] has a solution.
2. Explain why the columns of A are linearly dependent iff the system [A | 0] has a nontrivial solution.
3. Prove Theorem 3.4(d).
4. Prove Theorems 3.6 and 3.9(a-d).
   
5. Prove Theorem 3.19.
6. Explain why if A is invertible then Ax=b has a unique solution.
7. Explain why if A is invertible then Ax=0 has only the trivial solution.
8. Explain why if A is invertible and has n columns then col(A) is R^n.
9. Explain why A is invertible iff its columns are linearly independent