- #1

Winzer

- 598

- 0

## Homework Statement

I want to make sure I understand this.

True False.

1. Every matrix is row equivalent to a unique matrix in echelon form.

2. If u,v, and w are nonzero vectors in R2 then w is a linear combination of u and v.

3. If a system of linear equations has no free variables, then it has a unique solution.

4. If a system of linear equations has two different solutions, it must have infinitely many solutions.

5. If matrices A and B are row equivalent, they have the same reduced echelon form.

6. If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.

7. A linear transformation is a function.

8. If a system Ax = b has more than one solution then so does the system Ax = 0.

Question 8 answers

9.The equation Ax = 0 has the trivial solution if and only if there are no free variables.

10. If {u, v, w} is linearly independent, then u, v, and w are not in R2.

11.If BC = BD then C = D. For matrix

12. If AB = I, then A is invertible

13. If AB = BA and A is invertible, then A^-1*B = B*A^-1.

14.If b = [1 0 0]^T (i.e b is a column vector) and if A is a 3 x 3 matrix such that Ax = b has a unique solution, then A is invertible.

15.If AB = C and C has 2 columns, then A has 2 columns.

## The Attempt at a Solution

1. True.

2. False. This is not true all the time.

3. False. It could be inconsistent with a pivot in last column.

4. True. Linear dependence?

5. True. Because two matrices are equivalent if they have the same solution set.

6. False. Not too sure on this one. Isn't an identity matrix formed by AC=CA=I(identity matrix)

7. True. Ax=b=T(x)

8.True.

9. True. Do they mean a trivial solution or solutions.?

10. False. At least one of them could be in R^2.

11. False. Have a feeling because multiplication properties of matrices are not the same the other way around.

12. True. Because A is invertable if and only if A is row equivalent to I(identity matrix).

13. True. Can be reduced to identity matrix.

14. True. Got a hunch that it is true, but not completely sure that it is true.

15. True. Multiplication rules.

Please tell me my wrong doings and tell me why please. Thanks.