True or false. 1. You do not need to do the backward phase of Gauss-Jordan elimi
ID: 3028270 • Letter: T
Question
True or false.
1. You do not need to do the backward phase of Gauss-Jordan elimination to nd the pivot positions of a matrix.
2. If A is an m × n matrix and its columns are linearly independent, then m n.
3. Some element of a linearly dependent subset of a vector space is a linear combination of the other elements.
4. The columns of an m × n matrix A form a basis for Rm m = n.
5. A linearly independent subset of a vector space cannot contain 0.
6. A subset X of a vector space is a subspace if and only if X = Span(X).
7. If a vector space has an nite basis, then all of its linearly independent subsets are nite.
8. Every spanning subset of a vector space contains a basis.
9. A matrix can be row equivalent to more than one matrix in REF.
10. Equivalent linear systems have row equivalent augmented matrices.
11. Two matrices are row equivalent they have the same RREF.
12. When performing row operations, you are allowed to multiply a row by any real number.
13. All row operations are reversible.
2
14. Free variables correspond to pivot columns in the coecient matrix.
15. An m × n augmented matrix has n pivots its system is inconsistent.
16. Ax = 0 has a nontrivial solution every column of A has a pivot.
17. Ax = b is consistent its solution set is a translation of Nul(A).
18. Every subspace of Rm contains the origin.
19. A matrix equation Ax = b is homogeneous x = 0 is a solution.
20. Matrices A and B are row equivalent there is a row operation that transforms A into B.
21. Span() = .
22. A basis is a minimal spanning subset of a vector space.
23. The columns of an m × n matrix A span Rm A has n pivots.
24. Linearly dependent subsets of vector spaces are always nonempty.
25. A vector space is nite-dimensional all of its linearly independent sub-sets are nite.
Explanation / Answer
1) False, The final step is to back-substitute the solution already obtained for the 1 unknown into the previous equations to find the values of all the other unknowns.
2) True, • if an m × n matrix has linearly independent columns then m n • if an m × n matrix has linearly independent rows then m n
3) True
4) True
5) True
6) False
7) False , If span(v1, . . ., vm) = V , we say that (v1, . . ., vm) spans V . The vector space V is called finite- dimensional, if it is spanned by a finite list of vectors. A vector space V that is not finite-dimensional is called infinite-dimensional.
8) True
9) False
10) True
11) True
12) True
13) True, You can reverse multiplying by a constant by multiplying by its inverse. If you add row one to row two and replace row two, then you can subtract row one from row two to get it back.
14) True
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